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About Vijilant
(Top Expert on this page)

Expertise
Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

Experience
Teacher of math for 50 years

Organizations
ATL

Publications
Journal of mathematics and its applications

Education/Credentials
BSc Hons Liverpool

Awards and Honors
State Scholarship 1955

Past/Present Clients
I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.

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Questions Answered By Expert Vijilant
In Category Number Theory

Subject	Date Asked

Number Theory - Mersenne Primes	11/16/2009
Q: In a chapter titled Mersenne Primes, a question is asked not of the Mersenne prime format, and I am ... A: This question is really about the factor theorem. Do you remember it? A polynomial p(x) has the ...
ap calculus ab	11/10/2009
Q: my problems (set 2): In Exercise 19 find dy/dx. 19. y=(x+1)(x+2)/(x-1)(x-2) 20. Use the definition ... A: I hope these aren't homework problems. It's OK if you have managed to do 1 to 18 yourself. 19. You ...
ap calculus ab	11/8/2009
Q: my problems (set 1): 2. What is the limit of (4-x²)/(x²-1) as x approaches ∞? (A) 1, (B) 0, ... A: These aren't problems but routine examples. I suspect you haven't studied worked examples of the ...
ap calculus ab	11/4/2009
Q: my problems (set 2): For Exercises 9 and 10 find the derivative of each expression and simplify. 9. ... A: You don't say what to differentiate with respect to, but I shall assume it is x. 9. (1/a)(2x/b - ...
urgent math	11/3/2009
Q: Hey there I have a test tomorrow nigh for Calc and I just can't seem to get the answers for these ... A: You have left it late. Why didn't you query days ago. It is past midnight here in the UK and ...
trigonometry	10/31/2009
Q: Sir, please solve these problems soon, If cos2B=cos(A+C)/cos(A-C) then show that tanA,tanB and ... A: This is not my expertise, but I will help you with some of them. cos(2B) = 2 cos^2(B) - 1 = ...
math	10/27/2009
Q: i dont get how you figure out how far away a term is from another example: 1,2,6 ,120, ,5040 A: Recognising a sequence is a matter of knowing the rules of basic sequences and trying to see if you ...
Number Theory	10/26/2009
Q: if n>4 is a composite number, show that n\|(n-1)! Conclude that (n-1)! is not congruent to -1(mod n). A: Unless n is a square, it can be factorised, and both of these factors are less than n, and can be ...
Number Theory	10/26/2009
Q: I will use = to mean congruent. Using the fact that p-j = -j(mod p), show that if p is an odd ... A: Sometimes substituting in an answer you are asked to prove helps, so you were right to try that. ...
Number theory	10/21/2009
Q: Sir, 1) How many pairs of non-negative integers (m,n) each sum to 1492 without any carries? 2) If ... A: 1) Call the two numbers A and B. They are not equal since that would involve carrying. Number A has ...
Circles	10/21/2009
Q: I need ur help though my question is not from number theory but geometry I1, I2........ I2005 are ... A: You certainly ask some varied questions! The only geometry needed here is 'arc = rǾ.' So Ik has ...
urgent math	10/21/2009
Q: I REALLY need to know how to go about finding the inverse of a transcendental function. I have no ... A: Don't worry over this. You aren't going to be asked to do something impossible. Trust me. Inverse ...
Math	10/19/2009
Q: I just don't know where to start on these questions just some advanced questions that I didn't have ... A: The given information says \|x-2\| < δ, so multiply by 2 to get \|2x-4\| < 2δ. then δ = ...
Number Theory	10/19/2009
Q: find for what values of n "phi(n) is odd". Thanks for your help ANSWER: Hello Skrosuri The only ... A: 1. It is easily proved that the φ(n) integers less than n and prime to it form a ...
Compex Numbers	10/18/2009
Q: Can Sinus and Cosine work on complex numbers?(For instance can we have Cos(i)?) Many Thanks ... A: This is not really my expertise, but I'll answer it. Of course it is necessary to redefine sin and ...
Number Theory	10/16/2009
Q: Prove that when n is odd and a is an integer that a^n = a(mod3), where '=' means congruent. My ... A: This is another that can be done using FLT. Let n = 2k + 1. Then a^2 = 1 (mod 3) if a is not a ...
Number Theory	10/16/2009
Q: Show that for any integer, a, a^561 = a(mod 561), with '=' meaning congruent. My thoughts: Since ... A: A composite number n such as 561 with a^n = a (mod n) for all a is called a Carmichael number, and ...
mobius inversion	10/15/2009
Q: let f be an arithmetic function and let g(n) = ∑ (over d\|n) of μ(d)f(n/d) show that f(n) ... A: This is a standard theorem in the theory of arithmetic functions and the proof will be found in most ...
math -sequencing-arithmetic/geometric	10/14/2009
Q: what is the missing term in the following sequence ___,1_____,______,125,______,... ... A: With the usual notation for an AP, a + d = 1 and a + 4d = 125 Subtracting gives 3d = 124 and d = 41 ...
Riemann Hypothesis and Prime Numbers	10/11/2009
Q: Can you simplify and explain the Riemann Hypothesis up to a middle-school math level? A: The answer is a very firm NO. I have the book Prime Obsession by John Derbyshire, which I think is ...
Number Theory	10/11/2009
Q: find for what values of n "phi(n) is odd". Thanks for your help ANSWER: Hello Skrosuri The only ... A: 1) This is just another way of stating the Chinese Remainder theorem. suppose (mi,mj) = h. Then mi ...
Number Theory	10/10/2009
Q: find for what values of n "phi(n) is odd". Thanks for your help A: The only values of n for which phi(n) = 1 are 1 and 2. Suppose in the prime factorisation of n, a ...
Circles	10/8/2009
Q: Given two concentric circle of radii r,R From a point P on the smaller circle, a straight line is ... A: This certainly isn't number theory, so I should reject it, but I'm pretty good at geometry, so I'll ...
number theory	10/2/2009
Q: Q 1: A= (1999)^1999 B = sum of digits of A C = sum of digits of B D = sum of digits of C Find sum ... A: Q1 1999 = 1 (mod 9) So 1999^1999 = 1 mod 9. log(A) = 19998log(1999) = 6598.3.... This means A has ...
math	9/26/2009
Q: This is question from absolute value properties Question \|a\|>= and \|a\|>=-a Proof: if a>=0 then ... A: I just think you need to look at a few examples, or look at a number line. a = 3. \|a\| = 3. -a = ...
division	9/26/2009
Q: If we divide a number, is the answer always smaller? A: I presume we are talking about positive numbers here. Dividing a number by a number greater than 1 ...
ap calculus preparation	9/24/2009
Q: 9. For question 9, find the standard equation of the circle satisfying the given conditions. A ... A: I should reject these questions since (a) These are not number theory questions, but coordinate ...
Number Theory	9/22/2009
Q: I'll use = to mean congruent.... if a=b(mod n) and d\|n, show a=b(mod d). I assume that I would ... A: An exam question might give you the clue; prove from the definition of congruence... So in this sort ...
number theory	9/21/2009
Q: My questions are : a>How many triangular number ,less than 1000 ,can be expressed as the difference ... A: (a) The difference of squares of consecutive natural numbers is calculated as (a+1)^2 - a^2 = 2a + ...
Number Theory - Congruences	9/17/2009
Q: I have 3 questions. 1) 1) Show that for infinitely many n, 43\|(n^2 + n + 41). The answer will be ... A: We have to find solutions to the congruence equation: n^2 + n +41 = 0 mod 43 41 = -2 (mod 43) so (n ...
number series	9/1/2009
Q: find out the missing number: 8,12,18,_____,40(1/2) A: I'll assume you mean 40.5 for the last term. In that case it is a geometric progression. You get ...
4th year university number theory question	8/30/2009
Q: I need to show that the gcd(lcm(a,b), lcm(b,c), lcm(c,d)) = lcm(gcd(a,b), gcd(b,c), gcd(c,d)) is ... A: No wonder you had difficulty proving it! It isn't correct. Take any prime p, and denote the index ...
number theory proof	8/27/2009
Q: I need to show that the sum of the reciprocals 1+1/2+1/3+...+1/n is never an integer for n>1. I've ... A: I'm not going to set out the whole proof, but give you the idea of how to proceed. You can use ...
number theory	8/27/2009
Q: I need help with the following problem. I can't figure out how to set it up. (Brahmagupta, 7th ... A: I didn't give the smallest solution first time. Here is the revised version. This is a fairly ...
number theory2	8/22/2009
Q: look the image please A: I'll answer your first question and think about your second. Every prime number of the form 4k + 1 ...
math	8/12/2009
Q: Let a and b be two non zero rational numbers such that the equation ax2+by2=0has a non zero ... A: From the given equation, putting the solution x = c/d, y = e/f, c,d,e,f integral, we arrive at a = ...
inequalities	7/29/2009
Q: if a,b,c,d>0 show that a/b + b/c + c/a >= 3 and a^2 + b^2 + c^2 >= 3.(a^2.b^2.c^2)^1/3 A: These are very easily proved by using the theorem that for a set of positive numbers, their ...
Interest & Percentages	7/28/2009
Q: An investor wants to divide $1000.00 into two amounts, $800.00 and $200.00 for a total return of ... A: You are still an algebraphobe then. There isn't. If we do use algebra 200 at x% and 800 at y% we ...
Math	7/27/2009
Q: composite integers n for n divides 2^(n)-2 are called pseudoprimes ,Show that every fermat's Number ... A: It's a good job there was a hint. I could have been hours working this one out without it. Fn = ...
clarification	7/27/2009
Q: Your question is not clear, Does x3 stand for x^3 and x2 stand for x^2. Does 1/3x mean (1/3)x or ... A: After multiplying by 27, you get f(x) = 27x^3 + 27x^2 - 72x + 28 = 0. It is now a matter of using ...
finitude of primes	7/27/2009
Q: Vijilant, How do I prove that there is not a finite number of primes of the form 4k + 3 and 4k - 3, ... A: These are theorems in most textbooks. I'll give you an outline, but you should find the proofs in a ...
number theory - prime numbers/perfect numbers	7/25/2009
Q: Could you help me prove that if 2n-1 is prime, then 2p-1(2p – 1) is a perfect number? (Perfect ... A: A lovely name. It was my mother's. You seem to have a garbled version of the theorem. It should ...
number theory	7/24/2009
Q: for any integer 'n' greater than 1,prove that (4^n+n^4)is never a prime. A: When n is even, the number is even. When n is odd and prime to 5, 4^n is congruent to -1 mod 5, and ...
number theory	7/22/2009
Q: in 1886, a 16 year old italian boy announced that 1184 =2^5.37 and 1210=2.5.11^2 form an amicable ... A: This is question 10 of problems 10.2 of Burton's book. No answer is given. We need to show that ...
number theory	7/22/2009
Q: prove that the Mersenne number M19 is prime hence the integer n=2^18(2^19-1)is perfect Hint:the only ... A: There is a theorem that says that the only divisors of M(p) where p is an odd prime have the form ...
number theory	7/22/2009
Q: Verify: The product of two odd primes is never a perfect number? Hint: expand the inequality ... A: I think you can do this one for yourself. You know that sigma(pq) = sigma(p)*sigma(q) = (p-1)(q-1) ...
number theory	7/22/2009
Q: Verify that: a) no power of a prime can be perfect number. b) A perfect square can't be a perfect ... A: I hope you are doing similar questions for yourself. I don't just want to do your homework for you. ...
number theory	7/22/2009
Q: The Ciphertext message produced by the RSA algorithim with key(n,k)=(1643,223) is : 0833 0823 ... A: 1643 = 3153, so we need the inverse of 223 (mod 3052) 7223 = 1561 = 3052 + 1, so this checks ...
Number theory	7/21/2009
Q: show that 3 is a quadratic residue of 23 , but a nonresidue of 31 A: You need to be familiar with the law of quadratic reciprocity to do this one. (p/q)*(q/p) (where ...
prime-perfect square	7/21/2009
Q: I have two questions! No prime number can be a perfect square?Which of the following is not a ... A: The tower of Hanoi is certainly a math puzzle. Tetris and John Conway's Game of life are perhaps ...
number theory	7/21/2009
Q: if every prime that divides n also divides m ,establish that Φ(nm)=n Φ(m), in patricular ... A: Let the prime divisors of m be p1,p2,p3,....pr. Then nm has the same set of prime divisors. Then ...
number theory	7/21/2009
Q: prove that if the integern has r distinct odd prime factors, then 2^r divides Φ (n) A: Nothing is said about n having some power of 2 in its representation, but if it has the factor 2^k, ...
nimber theory	7/21/2009
Q: verify that the equality Φ(n)= Φ(n+1)= Φ(n+2) holds when n=5186. where Φ is ... A: We need the prime factorisations of each number. Then we use the multiplicative property. ...
number theory	7/16/2009
Q: if 'a'is prime to 'b'then (a+b)and (a^a-a^b+b^b)has no common factor other than '3'.-prove it. A: I think you may have mistyped this. I think the second number should be a^2 - ab + b^2. If this is ...
Maths	7/15/2009
Q: Why does 0! = 1 ? A: It is a matter of definition. But a sensible one. It is clear that n! = n*(n-1)! for n > 1. The ...
RE : Again on modular algebra problem	7/6/2009
Q: just to begin thanks a lot for your prompt answer for my past post. In fact maybe I explained the ... A: Sorry about the delay. I've had a busy marking load. If you can select the elements of S, it is ...
pascal's triangle	7/6/2009
Q: I don't understand this.... Find a relationship between entries in Pascal's triangle and the maximum ... A: First we need to find a formula for the number of diagonals of an n-gon. nC2 =n(n-1)/2 gives the ...
Possible ways of eliminating for a prime number	6/11/2009
Q: I am writing a JAVA program to find the next largest Mersenne's prime number. The standard way of ... A: This is an impossibly difficult question. On September 6th, the 46th known Mersenne prime, ...
prime numbers and divisibility	6/8/2009
Q: Please help. I know that this may seem so elementary but these problems really have me stumped. ... A: The sum you require is 3+9+15+...999. That is, the odd multiples of 3. It is an arithmetic series ...
logarithms	6/3/2009
Q: 1. If log_12(27)= a prove that log_6(16) = (4(3-a))/(3+a) 2. Let r = log_b(8/45) and s = ... A: Your tutor must be a sadist. These are the trickiest problems I have ever seen on logs. Perhaps ...
hi sir pls help in this question	5/31/2009
Q: a number consist of two digits . the digits in tens place exceeds the digits in units place by 4. ... A: Take x as the units digit; then the ten's digit is x+4. The value of the number is then 10(x+4) + x ...
Number Theory: Sums of 2 squares	5/26/2009
Q: Prove that if p is a prime number with p = 1 (mod 4) then p^2 = c^2+d^2 for some integers c and d ... A: First we need to prove or assume a theorem of Fermat which states that if p is a prime of the form ...
Number puzzle	5/18/2009
Q: Using each of the numbers 1 to 9 exactly once to fill in the blanks and complete this equation: ... A: I don't normally answer number puzzles, but since my wife is Sheila, I'll make an exception. 12786 - ...
squaring in base 6	5/12/2009
Q: The question I have is after converting 95 to base 6, the answer is 235. How do I square 235 in ... A: The calculation is the sum of 9 multiplications. 200200 = 40000 20030 = 10000 (23) = 10 2005 ...
Multiple Investment Accounts	5/11/2009
Q: Some time ago I asked the following question and you determined the answer by using the solution ... A: You won't let this one go will you. Let's start with one of the solutions for the 3 account problem. ...
number theory	5/10/2009
Q: How do I prove, If n^2 + m and n^2-m are perfect squares then m is divisible by 24? A: I can give a solution of this problem, but there may be a much simpler one. n^2 + m = a^2, n^2 - m = ...
fibonacci - prove by induction	5/7/2009
Q: Prove f(fibonacci)4n is a multiple of three. My approach has been to make f4k+1=3k+3... But I am ... A: To do this you need to do a little investigation. Look at the pattern (mod 3) of the fibonacci ...
proof by induction	5/6/2009
Q: not sure if you remember me but I am back :) We started learning proofs by induction. Could you ... A: I think you must be confusing the notation. <= is less than or equals. n! is never greater than ...
congruences mod n	5/5/2009
Q: I need to know all the cases (for different mods), how do you find the residue of a large number ... A: Greetings Tom There are many methods available. In the case of a composite power the simplest is to ...
proof by induction	5/5/2009
Q: not sure if you remember me but I am back :) We started learning proofs by induction. Could you ... A: 1. n = 1 LHS = 1^3 =1 RHS = 1^222/4 = 1, so theorem true for n = 1. Theorem true for n = k gives ...
number theory, least common multiple	4/29/2009
Q: my ten year old is having a hard time understanding it can you help me explain for example 5,11, and ... A: You have to find the smallest number which is a multiple of each of the numbers. Start with 5 and ...
discrete mathematics	4/18/2009
Q: I have been trying to solve a problem: prove or disprove: the sum of three irrational numbers is ... A: Given t irrational, to prove 3t is irrational. Suppose to the contrary, then 3t = a/b with a,b ...
discrete mathematics	4/18/2009
Q: I have been trying to solve a problem: prove or disprove: the sum of three irrational numbers is ... A: This is the sort of question, that once you've seen the answer, you could kick yourself for not ...
discrete mathematics	4/17/2009
Q: I have been trying to solve a problem: prove or disprove: the sum of three irrational numbers is ... A: Well done. You have done the hard part. The only thing that is not accurate is the initial ...
discrete mathematics	4/17/2009
Q: I have been trying to solve a problem: prove or disprove: the sum of three irrational numbers is ... A: I'm not sure I understand the question. But to prove r3 is irrational is easy. Suppose to the ...
discrete mathematics	4/17/2009
Q: I have been trying to solve a problem: prove or disprove: the sum of three irrational numbers is ... A: You are special because you have the same name as my mother born 1908. This is one you can disprove ...
sequences	4/15/2009
Q: I'm trying to learn how to find the 7th term in the following sequence... -4,-6,-8,...,-2n,-2,... I ... A: You have the answer in front of you, but you have put an extra comma where there should not have ...
Multiple Investment Accounts	4/9/2009
Q: Some time ago I asked the following question and you determined the answer by using the solution ... A: All the amounts must be positive. So 10,000 - 2t > 0, 20,000 -t > 0 and 3t > 0. From the first, 2t < ...
number theory	4/1/2009
Q: Prove or disprove if (a^n) +1 is a prime number, than a is even and n is a prime number. also.. let ... A: For the first, we disprove by giving a counter-example. 1^1 + 1 = 2 is prime, but neither a =1 is ...
probability and combinations	4/1/2009
Q: The “Pythagorean Triple Threat” is a co-ed softball team. The league they are participating in says ... A: Not too well actually. For the first part you have calculated the number of ways of permuting ...
Number Theory	3/30/2009
Q: 1. Let a be any rational number that is not a perfect kth power. Show that the kth root of a is an ... A: I don't normally answer 4 different questions in one post. Today, you are lucky. I don't have ...
permutation and combination	3/28/2009
Q: A train going from Delhi to Jaipur stops at seven intermediate stations.Five persons enter the train ... A: Perhaps that is the answer to a different question. There is no way the answer can be 10C2 - 10. ...
permutation and combination	3/26/2009
Q: Suppose N different games are to be given to N children.In how many ways can this be done so that ... A: Unlike your previous question, this is a very easy one. Give all the children one game. This can be ...
permutation and combination	3/26/2009
Q: A train going from Delhi to Jaipur stops at seven intermediate stations.Five persons enter the train ... A: I am assuming that we are only concerned with which tickets there are, and not with the order in ...
permutation and combination	3/26/2009
Q: A die is rolled ten times.Find the number of ways so that the outcomesalways contain 1,2,3. A: I am assuming that by outcome you are only concerned with the count of each digit that ensues, not ...
Number Theory	3/19/2009
Q: How do I solve: Show that if d\|m then phi(d)\|phi(m). Note: phi(m),phi(n) is Euler's phi function. ... A: With a question like this, a lot depends on which theorems you are allowed to assume. The obvious ...
Congruences/methods of calculation	3/19/2009
Q: The question is: Show that 1^2, 2^2,...,m^2 is not a complete residue system modulo m if m > 2. ... A: Your question has now moved from fresher to graduate school. I did study pseudoprimes about 45 years ...
pythagorean tripples	3/18/2009
Q: Doing a section on Pythagorean tripples, and I understand the simple examples, but these I just ... A: I'll show you a method for the first one which should enable you to do the second yourself. As with ...
Mills & Decimals	3/17/2009
Q: I read the following in an old business mathematics textbook: "Express 13 mills as a decimal part ... A: frac() is just the same as decimal part of(). In other words you throw away the integer part of the ...
Number Theory	3/16/2009
Q: find 13 consecutive composite positive integers...is there a formula for finding this? Prove that ... A: Consider 14! + 2, 14! + 3.....14! + 14. The first has factor 2, the second factor 3... the 14th ...
Number Theory	3/16/2009
Q: Pn = P1P2P3….Pn +1 , where the Pi’s are the primes in increasing order of value. Prove that Pn is of ... A: We are concerned here about divisibility by 4 and 3. So we need to use the fact that P1 =2 and P2 = ...
Mills & Decimals	3/15/2009
Q: I read the following in an old business mathematics textbook: "Express 13 mills as a decimal part ... A: You must be researching the history of maths teaching. The decimal part of a number is the part of ...
recursion problem	3/3/2009
Q: a1=3, ak+1=ak-2 what is the value of the 4th term 1 is below a, k+1 is below a, and k is below a A: I was at the gym on the treadmill this morning, and sometimes that is when I have my best thoughts. ...
number theory	3/3/2009
Q: (a U c) U (b U c) how to distribute and what is the answer I believe you get, a U b U c U A: First of all consider a Venn diagram. Three intersecting sets. The union of two sets can be ...
additive/multiplicative inverse	3/2/2009
Q: When does a have an additive inverse (mod m), when does a have a multiplicative inverse and how ... A: a always have an additive inverse (mod m). It is -a, or m-a if you need a positive residue. a has a ...
(A xor B)+ (B xor C)+ (C xor A)=(A xor B)+ (B xor C)	2/18/2009
Q: How can I prove that (A xor B)+ (B xor C)+ (C xor A)=(A xor B)+ (B xor C) using algebraic ... A: This is not quite my field of expertise, but I used to teach computer science, so I can answer the ...
polynomial congruences	2/6/2009
Q: This problem is really throwing me off, I keep playing around with it, but I'm obviously not doing ... A: An interesting one which I haven't seen before. First of all (p-1)! = -1 (mod p) by Wilson's ...
Congruences	2/4/2009
Q: The question is: Show that 1^2, 2^2,...,m^2 is not a complete residue system modulo m if m > 2. ... A: Let me answer your second question first. When we are asked to disprove something as in this case, ...
binomials	1/31/2009
Q: What is the binomial series for (a + x)^63 thanks A: Not even mathcad can display all 64 terms of this expansion, but I can give you the first few terms, ...
Fermat's Last Theorem	1/29/2009
Q: Find all integer solutions for a, b, c not congruent to 0 mod 5 for a^3+b^3=C^3 mod 5. I don't know ... A: I don't know how long this has been in the question pool. I don't usually look there. The question ...
euler phi function.	1/28/2009
Q: There are two parts, prove phi(2n)=phi(n) and prove phi(2n)=2phi(n). Is the best way to solve this ... A: The first case is when n is odd, so that the factor (1 - 1/2) does not occur in the calculation of ...
Simple Interest & Multiple Accounts	1/27/2009
Q: Some time ago I asked the following question and you determined the answer by using the solution ... A: It's because you can't have a negative amount of cash in an investment account. Unless you know ...
Simple Interest & Multiple Accounts	1/26/2009
Q: Some time ago I asked the following question and you determined the answer by using the solution ... A: Yes there are lots of answers, only limited by the fact that you would want to have the number of ...
Simple Interest & Multiple Accounts	1/24/2009
Q: Some time ago I asked the following question and you determined the answer by using the solution ... A: You already have a solution. 10,000 in the 2%, 0 in the 3% and 20,000 in the 5%. To put some in the ...
number theory	1/22/2009
Q: I have two number theory questions. First, show that 561 is a pseudoprime, where 561=3(11)(17). ... A: I'll start with the second because that is much easier. The FLT assures us that a^(p-1) = 1 mod p ...
Calendar Weeks & Days	1/19/2009
Q: If someone is paid every two weeks, he receives 26 paychecks per year, (52 weeks/year)divided by (2 ... A: Your original assumption that there are 26 paydays in a year is false. If payday is a Friday and ...
Number theory/divisibility	1/18/2009
Q: My main question is this, since I'm a bit rusty: Let s and g>0 be given integers. Prove that ... A: First if and only if. Propostion A if and only if Proposition B To prove this we have to prove A ...
proof	1/17/2009
Q: I have to prove by Contradiction the following If x,y are real numbers such that x + y is ... A: The idea of a proof by contradiction is to start by supposing the opposite of what we have to prove ...
Set theory	1/8/2009
Q: simplify, showing your working out A n (A U B) ANSWER: Hello Mohammed We use the distributive law ... A: I make a rule not to answer more than 3 questions in one post. That is because i believe the student ...
absoulte value	12/28/2008
Q: \|a\|≥a and \|a\|≥a Proof if a≥0 then \|a\|=a If a<0 then \|a\|=-a>a aєR ... A: An example \|3\| = 3, so \|3\| >= 3 \|-5\| = -(-5) = 5, so \|-5\| > -5 I think maybe you thought -a ...
maths ask a question	12/18/2008
Q: is there any formula to find the sum of squares of the terms of an arithamatic progression ? A: Amazingly, I have never seen this question before, but it should be easy enough. S(n) = a^2 + ...
Number Theory	12/5/2008
Q: I have 3 questions. 1) Given a is an element of Z, Prove: If a is even, then a^2=0(mod4) and if a ... A: 1) If a is even, the we can write a = 2b. Then a^2 = 4b^2 = 0 mod 4. If a is odd, we can write a = ...
Permutations and Number Combinations	12/3/2008
Q: Using your example 4 placeholders out of 20 places and getting the answer 4845 number combniations, ... A: I'm not sure I understand what you mean here. Suppose we represent the 20 objects by the first 20 ...
number theory	12/3/2008
Q: prove the converse of Wilson's theorem A: The converse is. If (n-1)! = -1 mod n, then n is prime. If n is not a prime, then it has a divisor, ...
Mersenne primes and perfect numbers	12/2/2008
Q: A) show that a power of 3 can never be a perfect number B)more generally, if p is an odd prime, show ... A: First of all, your formula is not correct, it has an incorrect sign. A) Suppose it can, ...
Number theory	12/1/2008
Q: If a^n+1 is a prime for some numbers a greater than or equal to 2 and n greater than or equal to 1, ... A: I don't think your statement a^n+1=a^m2^p+1=a^2pm+1=b^m+1 is correct. It doesn't include any ...
Follow up question	11/18/2008
Q: I just had a few follow up questions about a question that you previously answered. Here is the ... A: x^2 + 5x + 1 = 0 mod 47 5 is even so add 47 to get x^2 + 52x + 1 = 0 mod 47 Now you can complete the ...
Number Theory	11/18/2008
Q: Find the number of solutions to the congruence equation x^2+5x+1 ≡ 0 mod 177425. Note that ... A: I think this is a copy of the answer you need Questioner: Jules Category: Number Theory Private: ...
mathmatics-number theory	11/17/2008
Q: determine those odd primes t for which -3 is a quadratic residue and those for which it is a non ... A: I assume that you can use the quadratic residues theorem. Firstly 2 is a a special case. -3 = 1 mod ...
recurring decimals	11/16/2008
Q: quite often, but not always, the digits that form a recurring decimal occur in complimentary halves: ... A: This is a very interesting question, and I hadn't realised that before, so thank you. I will try to ...
heavyweight	11/14/2008
Q: i am wondering if you could answer this for me.how much in, simplified form,is this:1e+60.like how ... A: I wonder if you are the Rodney from 'All fools and horses'. Or should it be elephants? 1E60 is 1 ...
number theory	11/12/2008
Q: 1. Prove that ∛5 is irrational. 2. For any numbers a, b, (a, b)ⁿ = aⁿ ... A: I don't normally answer such a lot of questions in one post. Please don't ask more than 3 in any ...
Series - Arithmetic or Geometric?	11/11/2008
Q: We know that the two following series is arithmetic and d=2. 2, 4, 6, 8, 10 3, 5, 7, 9, 11. ... A: A series in which each term is the reciprocal of the corresponding term in an arithmetic series is ...
Modular arithmetic	11/4/2008
Q: How do I determine the last 2 digits of 3^400 using mod 100? ANSWER: Hello olebile phi(100) = ... A: I thought you might have been able to do #1 if you studied the previous answer, because we use the ...
hard math problem	11/3/2008
Q: Find, with proof, the largest positive integer k with the following property: There exists a ... A: I don't know that there is a largest one. But let us take an example that works, and see why. ...
Number Theory	11/3/2008
Q: What is the last 2 digits of 123^456? A: First of all, we need 123^456 mod 100. We use Euler's generalisation of Fermat's little theorem. ...
Fractions	11/1/2008
Q: Example 1. 2/3 of Mary's age and 3/4 of Ruth's age equal Sarah's age. If the total ages equal 46, ... A: I'm sorry to disappoint you, but your method does not give the correct answer in the example. 2/3 of ...
modular proof	9/29/2008
Q: i am having a real difficulty in proving this statement by induction. == is congruent symbol d> or ... A: This could be proved directly using the binomial theorem, but if you need a proof by induction here ...
Number theory Proof	9/26/2008
Q: Let m and k be positive integers. Then the number of positive integers less than or equal to m*k ... A: My most sincere apologies for my horrendous mistake. I've answered so many questions successfully ...
a counting and enumeration problem	9/24/2008
Q: I have a question that maybe I could have answered when I was taking probability 19 years ago but am ... A: This is an interesting problem I can help you with. Let us look at an example. How many selections ...
Number theory Proof	9/24/2008
Q: Let m and k be positive integers. Then the number of positive integers less than or equal to mk ... A: This is not true as stated. A counter-example is m=6, k=4. phi(6) = 4 so that kphi(m) = 4*4 = 16. ...
Odd and Even Multiples	9/21/2008
Q: I must find out how many multiples of 3 there are from 1-2008, then state how many and odd and even. ... A: This sounds like some Maths coursework where you have to put some graft in. Carefully list your ...
how to find the primes in just a few seconds	9/19/2008
Q: i just want to ask how to find the 3 largest primes of 555,555,555,555.I know it's 37, 9901 and 101. A: Your number certainly has the factor 555. It is 555(10^9+10^6+10^3+1) = 5337(10^6+1)(10^3+1) ...
Number Theory	9/18/2008
Q: I'm so confused, I have no idea how to solve these... can you answer them for me? It would be a ... A: 1) You need to use the Euclidean algorithm property. (a,b) = (a, b-ka) or (a, ka-b). here we use a = ...
Number Theory	9/17/2008
Q: I am new to this class and have never taken a proof class. I dont even know where to start on these. ... A: I'm not going to do all of these for you. They can all be proved by using the unique prime ...
maths	9/14/2008
Q: if A equals 1 and Z equals 26 what word when times adds up to 1 million A: I can only help you with the maths of this question. After that it is up to try different ...
terms and sequence	9/13/2008
Q: My daughter has a homework on terms and sequence with no real explanation of how or what they mean. ... A: A sequence is simply an ordered set of numbers, usually without end. e.g. 1,3,5,7,9,..... ...
Explain" Every number can be written as difference of two squares"	8/18/2008
Q: "Every number can be written as difference of two squares" For example 24=7^2-5^2, 32= ... A: Welcome to my answer service. The difference of two squares is factorisable. x^2 - y^2 = (x+y)(x-y) ...
number theory	8/17/2008
Q: 1.solve 6x+10y+15z=1 find x and y,z 2.if(a,4)=2,(b,4)=2,then prove that(a+b,4)=4; exibit three ... A: 4y+3z =1 mod 6 multiply by 2 8y + 6z = 2 mod 6 8y = 2 mod 6 1. Suppose there are only a finite ...
number theory	8/16/2008
Q: 1.solve 6x+10y+15z=1 find x and y,z 2.if(a,4)=2,(b,4)=2,then prove that(a+b,4)=4; exibit three ... A: You are asking a lot of questions. I hope you are trying to solve them yourself before you ask me. ...
number theory	8/15/2008
Q: 1.for every positive integer k show that (5^2)^k congruent to 1+3*2^(k+2) 2.let Fn=((2^2)^n)+1 Nth ... A: 1. You asked this earlier and I said it was impossible to answer, as you haven't specified a ...
number theory	8/15/2008
Q: 1.prove that product of k consiquetive intigers is divisable by k1! 2.for every positive integer k ... A: 1. Let the largest of the integers be n. Then nCk, the number of ways to choose k objects from n is ...
number theory	8/11/2008
Q: 1.find last two digits of 2^1000 2.(a,a+k)/k for a,knot equel to 0 3.(a,b)=1 and c>0 prove that an ... A: 1. 2^10 = 1024 =24 mod 1000 2^20 = 24^2 = 576 = 76 = -24 mod 1000 2^30 = -576 = 24 mod 100 Odd ...
prime	8/10/2008
Q: Find all prime numbers(p,q,r) , that numbers pq+pr+rq and p^3+q^3+r^3-2pqr are divided by p+q+r. A: I'm afraid I don't have the time at present to do justice to this problem, but I would not be ...
number theory	8/10/2008
Q: 1.find the no of solutions of x^2congruent x mod m and for all positive integers of m 2.pi(m)=pi(mn) ... A: 1. x(x-1) congruent to 0 mod m for every m, can only have the solutions x=0 and x=1. 2. I'm ...
Watts & Watt-Hours	8/8/2008
Q: Can you explain the difference between watt-hours per hour and watts per hour? I thank you for any ... A: Suppose we take the example of a 1 watt indicator lamp. It is using energy at the rate of 1 joule ...
combos	8/6/2008
Q: using 0-9 how many combinations are possible in a 9 digit combination A: Let's put it into a real situation. You are the manager of a team which plays 9 a side martian ...
find the remainder when 7^51 is divided by 144	8/4/2008
Q: find the remainder when 7^51 is divided by 144 A: We can use Euler's generalisation of Fermat's little theorem: a^phi(n) = 1 mod n if a is prime to n ...
(a,p)=1,(n,p-1)=1 then x^n congruent a mod p has exactly one solution	8/2/2008
Q: (a,p)=1,(n,p-1)=1 then x^n congruent a mod p has exactly one solution A: I will have to assume that you mean p is prime. The p-1 numbers, 1,2,...p-1 form a multiplicative ...
number theory	8/1/2008
Q: 1.x^3+4x+8 congruent to 0 mod 15 using chinese remainder theorem 2.how to solve 15x+12y+30z=24 A: We need to solve x^3 + 4x +8 = 0 mod 3 and also mod 5. The first is true for x=2 mod 3. The second ...
math	7/29/2008
Q: how many 6 digit combinations can you make from 17 to 47 EX- 17 19 20 22 33 47? and can you send me ... A: This is 31C6 as there are 31 to choose from and we need to choose 6. Most scientific calculators ...
plz enlighten me.	7/25/2008
Q: what about well ordering property,is there any proof in this? A: I think I said that the well ordering property is exactly that, a property and is part of the ...
Probability	7/24/2008
Q: What is the probability of (a) two randomly chosen sides and (b) two randomly angles being equal in ... A: There is a subtle difference here, as the consecutive property is now given. Since every pair of ...
Probability	7/24/2008
Q: What is the probability of (a) two randomly chosen sides and (b) two randomly angles being equal in ... A: (a) There are 3 different ways to choose the sides of the triangle, and only one is the equal sides, ...
im bothered!	7/21/2008
Q: how to prove well ordering principle? A: I don't think this is something you can prove. It is one of the properties of the set of natural ...
odd numbered fractions that dont equal 1	7/13/2008
Q: I was just running through some numbers and wondering how almost all odd numbered fractions don't ... A: Your calculator can't show a recurring decimal. 1/3 is 0.3 recurring. A reasonably modern ...
Number Theory	7/11/2008
Q: What is the formula to factor the binomial: 2^ab + 1 OR the binomial 2^ab - 1 (where a and b are ... A: If ab is a power of 2 there is no algebraic factorisation for 2^(ab) + 1 in real numbers. But we ...
number puzzle	6/25/2008
Q: Can you answer this: work out the logic of these numbers and state which other number completes the ... A: Something tells me that this is not really a maths problem. Your sequence can be written 119, 212, ...
number theory	6/3/2008
Q: I'm having trouble with this problem it's my first class in number theory so I don't have much of a ... A: Proofs in number theory are notoriously difficult, but don't despair. As you progress it gets ...
math	6/2/2008
Q: The width of a rectangle is half as long as the length.The rectangle has an area of 128 square ... A: We need to use some algebra and denote the width of the rectangle by w feet. Then the length of the ...
Can the following problem be solved using the Euclidean Algorithm?	5/15/2008
Q: This is a follow-up to my subject question about the local village gala The problem was solved for ... A: This has already been done. Here is a copy of my previous answer. We have two equations using pence. ...
6-digit combination	5/5/2008
Q: How can I get every possible 6-number combinations using numbers 0 to 49 on an excel spreadsheet? A: With great difficulty. You would need about 84 million cells. You could use a simple basic program ...
Last question in my sons homework.	5/1/2008
Q: My sons homework is number sequences. example 7n-3 = 4, 11, 18, 25 etc. I understand how this works. ... A: Sorry for the delay. I didn't get the email. My ISP is sometimes too careful about eliminating ...
proof	4/27/2008
Q: how can i prove that the number 1/square root of 2 is not a rational? how can i prove square of 2 is ... A: Sorry for the delay. I didn't get the email. My ISP is sometimes too careful about eliminating ...
Euclidean Algorithm	4/27/2008
Q: Can the following problem be solved using the Euclidean Algorithm? At a local village gala, the ... A: Sorry for the delay. I didn't get the email. My ISP is sometimes too careful about eliminating ...
Finding the nth term with no constant increase	4/10/2008
Q: I am stuck on finding nth terms when there doesn't appear to be a constant increase or decrease. is ... A: Sequences where there is a constant difference are called arithmetic sequences. Sequences where ...
Congruences	4/7/2008
Q: Prove that 111^333 + 333^111 is divisible by 7 A: We will look at each term separately. 111 = -1 mod 7 and -1 to an odd power is -1 333 = 4 mod 7 By ...
CRT	4/5/2008
Q: Sorry for the disturbance and I have the following confusion. Please clarify with suitable example. ... A: When we say there is one solution to a modulus equation, we mean that all the solutions are ...
Recurrene Relations	4/4/2008
Q: I know something and I'm looking for a proof of it. Consider this recurrence relation. ... A: You could show that the second order equation you have quoted does satisfy the recurrence relation ...
CRT	4/2/2008
Q: Sorry for the disturbance and I have the following confusion. Please clarify with suitable example. ... A: It all depends. If you have studied linear algebra, you will know that simultaneous equations can ...
divisbility	4/2/2008
Q: Please show me how to prove: if a\|c and b\|c and (a,b)=1, then ab\|c. Thank for your help. A: Probably the easiest way to see this is to consider the unique prime factorisation of a and b. The ...
Repeating Decimals	3/29/2008
Q: Can you think of another calculation, one that is more simple, that can determine the fraction for a ... A: When I was at primary school over 60 years ago, we just learned the rule to put the recurring digits ...
congurance	3/25/2008
Q: please see the following and give me a reply. generalize: Let a, b and c are some integers and c > ... A: You already know that you don't change the value of a mod b by adding a multiple of b. But cb is a ...
comgurances	3/24/2008
Q: I am mahima and I came with the following two question after long gap. Please help me and give ... A: The first one is just saying that adding or subtrac any multiple of b does not change the value of a ...
ZETA(4)	3/12/2008
Q: Is there any proof that shows us: ZETA(4)=pi^4/90 I can prove that ZETA(2)=pi^2/6 but for ZETA(4), ... A: You need to know how to calculate Fourier series. Define the function f(x) = -x for -PI<=x<0 and ...
in regaurds to your contropositive proof on 9/10/2007	3/7/2008
Q: I understand the basis for your contrapositive proof of n having to be a power of 2 for 2^(n) +1, I ... A: It is n that has to be a power of 2. If n has any odd factor e.g. n = s*t where s is odd then f(2^t) ...
sums of 2 addends of same parity	3/6/2008
Q: please tell me if you are familiar with this, or have reason to believe it holds or is false, ... A: I elected to answer this question, even though I have never come across this before. If I wasn't ...
possibilitys	3/5/2008
Q: what are all the possiblity of a 4 digit number using 0123456789 A: If repetition is allowed, each digit may be chosen in 10 ways, so there are 10^4 ways. If repetition ...
inverting fractions	3/4/2008
Q: why fractions have to be inverted to perform dividion? just my cureosety. A: This about inverses. How do you do the opposite of dividing by 2. You multiply by 2. It brings ...
Simple Solution	3/1/2008
Q: Here is myAn employee can retire from his place of employment if he meets the following ... A: Sorry for the delay. I didn't receive an email. My ISP is experimenting with a new method of spam ...
A Maths Mindbender...	2/7/2008
Q: There's a bus with 7 girls. Each girl carries 7 bags. Each bag carries 7 big cats. Each big cat ... A: Sorry for the delay. I didn't get the email. We will assume that the girls each have 2 legs. Girls ...
Pre Alg	2/5/2008
Q: Nth term My son was given the 2 problems to solve 1/7 2/9 3/11 4/13 came up with the expresion ... A: Sorrry for the delay. I didn't get the email. You have to notice what the pattern is. Each time ...
day of the week	2/3/2008
Q: January 1, 2023 will fall on what day of the week? A: If you just want an answer, the simplest way is to type the date 1/1/2023 in an excel or open office ...
Question on eye color gene	2/2/2008
Q: My mother has hazel eyes and my father has dark brown eyes. Out of 3 children they had 2 dark brown ... A: There is a problem about which genetic model of eye colour you are expected to use. The actual ...
infinity	1/29/2008
Q: I had two problems and these are still remains from last 3 years. one is: sigma 1/n where n tends ... A: Let me explain sum to infinity. No one is suggesting we add an infinite number of terms. But the ...
Number Theory	1/25/2008
Q: 1) Prove that if n>0 & (n^3)+1 is prime, then n=1 A: n^3 + 1 factorises as (n+1)(n^2 -n + 1) If it is prime, one of these factors must be 1. It is not ...
number theory	1/24/2008
Q: please see the following questions and answer and help me. 1. Show that the square root of a prime ... A: 1. Suppose to the contrary Then root(p) = a/b with (a,b) = 1 p = a^2/b^2 a^2 = p*b^2. Since (a,b) = ...
Set Theory Formulae	1/22/2008
Q: Could you please provide me with the formulas used in Set Theory..I'll be really grateful. Thanks ... A: I'm not going to reinvent the wheel. You should find everything you need to know at: ...
number theory	1/18/2008
Q: please kindly help me. 1. Let a1,a2,…,an be positive integers, if a1,a2,…,an are pair wise ... A: I presume the notation [ ] means lowest common multiple. When we multiply the numbers together, the ...
euclidean	1/18/2008
Q: Respected sir, I am navya. for ax = b (mod m)there is no solution if d does not divides b. where d = ... A: The equation can be written; ax = b + km a and m share the factor d =(a,m) so b must also have this ...
Difference	1/16/2008
Q: I had the doubt on the following: What is the exact difference between Proposition, theorem and ... A: A proposition is a mathematical statement such as "3 is greater than 4," "an infinite set exists," ...
Sequence	1/4/2008
Q: How to find the next term in a sequence? Is there a general rule? For example, the seqeunce ... A: The easy answer is that it could be any number you can think of. But usually we want the given ...
Rates	1/3/2008
Q: Which or what answer is mathematical correct in the following regarding these rates? One worker can ... A: Now I come to read your query again, I realise that there wasn't an original question. If the ...
fermat	1/3/2008
Q: I came to know that, the number of digits of a fermat number can be found by 1 + floor function of ... A: Here is a copy of a spreadsheet with F(n) worked out( only approximately for n>5), and the number of ...
number theory	12/31/2007
Q: What do you mean by "Sloane's A000045)." Is it person name or any thing? please giveme reply. ... A: Sloanes is an encyclopaedia of integer sequences, each one numbered. A sequence with such a low ...
gcd	12/30/2007
Q: I recently came to know for finding GCD of two numbers by the following formula. But I do not know ... A: This is just about impossible to explain without the ability to draw diagrams. The proof by Marcelo ...
fibonacci	12/30/2007
Q: How to prove that ' The sum of alternative Fibonacci numbers ending with F_k is (F_k+1)-1. Please ... A: The point is that your alternate terms end with F(k). If k is even, we need to start with F(2). For ...
fibonacci	12/28/2007
Q: I am mahima. 1,1,2,3,5,8,13,21,... is said to be fib series. here ratios are, 1/1 =1, 2/1 = 2, 3/2 = ... A: The golden ratio, phi, is the limit of the ratio of successive terms. It's value is (1+root5)/2 = ...
fibonacci	12/27/2007
Q: I am mahima with fibonacci problem. please answer or give some solution to go ahed of tjhe problem. ... A: I have produced a spreadsheet to help with the investigation. If you give me your email address, I ...
fibonacci	12/27/2007
Q: I am mahima with fibonacci problem. please answer or give some solution to go ahed of tjhe problem. ... A: I've already proved that the series is finite for base 2 and 3. I can't see any purely analytic ...
fibonacci	12/21/2007
Q: please see the following and answer. how about converting the Fibonacci numbers to a base other than ... A: This sounds like a long term project rather than a question. However, I can start you off on it. In ...
Convert from hexadecimal notation to binary notation	12/14/2007
Q: 1. 80E 2. 135AB 3. ABBA 4. DEFACED Please show work. Is this related to BASE calculations? If I can ... A: Each Hex digit is represented in binary by 4 binary digits. This is because 16 = 2^4. I am sure you ...
Solve the congruence	12/14/2007
Q: Solve the congruence 4x=5(mod9) 2x=7(mod17) There isn't a key on the computer for the triple (=) ... A: I am sorry if I didn't do all the explanation you needed, but I assumed you knew about the group ...
Interest	12/14/2007
Q: I have a question that I hope you can help solve. An investor wants to separate $30,000 into two ... A: I think I am getting quite good at no-algebra solutions. Suppose he invests all the cash at 2%. He ...
numbers	12/13/2007
Q: Sir, see the following: For every positive integer n, there are n consecutive integers that are ... A: Your formula is not the only way to find prime deserts. Your example starting at 90 shows that. ...
Solve the congruence	12/12/2007
Q: Solve the congruence 4x=5(mod9) 2x=7(mod17) There isn't a key on the computer for the triple (=) ... A: Have you met the Julia set in chaos theory? This can be solved by the Chinese remainder theorem, but ...
divisibility	12/11/2007
Q: We know that 1331 is divisible by 11. for conformation, we can also use that divisibility test 11. ... A: For divisibility by d, you used base d-1, and the 11's rule. I suggested that instead, you could use ...
divisibility	12/10/2007
Q: We know that 1331 is divisible by 11. for conformation, we can also use that divisibility test 11. ... A: You are using base d-1 to test for divisibility mod d. d-1 = -1 mod d (d-1)^2 = 1 mod d (d-1)^3 = -1 ...
Number Theory	12/5/2007
Q: Prove that n^4+1 cannot be a prime number. A: I think you have mistyped this. 2^4 + 1 = 17 is prime. So is 6^4 + 1 = 1297. What I think the ...
number theory	12/3/2007
Q: Sir, Please answer the below 2 questions. pl z. 1. Prove that there is no polynomial f(n)with ... A: 1. Suppose to the contrary. Then f(x) of degree n is prime for all integral x. Then f(0) = p where p ...
Justify	11/26/2007
Q: Sir, How can we justify that: In every set of ten different 2-digit positive integers, a)There ... A: I can help with the first one. a) This is proved by the pigeonhole principle. The maximum possible ...
Complex no. Problem	11/25/2007
Q: I came up with the following proof.Plz tell me whats wrong. Its here.(Consider i as iota) i^2 = -1 ... A: You have made a very common mistake, which many of my students do all the time. 'taking square root' ...
real analysis	11/25/2007
Q: what is/are the proofs that rational numbers are a field? ANSWER: Hello Brendz First we have to ... A: To be an ordered field, the field has to include a set P of positive elements which is closed under ...
N. Theory	11/24/2007
Q: Mr. Vijilant, I hope you are doing good and this is my last question for the year. If k = 2^50 ... A: This is not an easy question, and I don't believe it has an analytic answer since it deviates from ...
Twin prime separation	11/24/2007
Q: I am mahima and I came with the following questions. Please give me a comprehensive solution for the ... A: I don't think I can help you on this one. You don't say what n1, n2 etc are. From your formula it ...
High School Math	11/23/2007
Q: Sir, How are you doing? 1a) Our class of 16 people must elect a president, vice- president, ... A: This question is easy enough if you know the method. 1A We consider one possible combination of 24 ...
Problem Solving	11/23/2007
Q: Mr. Vijilant, what is wrong with the following problem situation? A survey of 144 new teachers ... A: Using E for Europe, S for Asia and F for Africa, and applying the Venn diagram law. n(EUSUF) = n(E) ...
High School Math	11/23/2007
Q: Sir, How are you doing? 1a) Our class of 16 people must elect a president, vice- president, ... A: I assume you are familiar with arrangements and selections. 1(a) The P can be elected in 14 ways. ...
Number Theory	11/17/2007
Q: Could you help me with the following question please? 1) find the number of four- digit numbers ... A: That looks like two more questions. I've only time for one tonight. I'll try to get back to you ...
Number Theory	11/17/2007
Q: Could you help me with the following question please? 1) find the number of four- digit numbers ... A: Sorry, I thought you would understand the notation. nPr denotes the number of permutations of n ...
Number Theory	11/14/2007
Q: Could you help me with the following question please? 1) find the number of four- digit numbers ... A: Sorry abou the delay. I didn't receive an email. (a) 5^4 = 625 (b) 5P4 = 120 (c) 3*5^3 = 375 (d) ...
induction question	11/14/2007
Q: having trouble with the proof. Prove that if n>=1 then, sqrt(1)+sqrt(2)+...+sqrt(n) > ... A: Sorry for the delay. I didn't get an email. I don't think this is one you can prove easily by ...
triplet	11/11/2007
Q: Sir, Thank you very much for my previous question. Now, please answer the following. Prove that ... A: I presume you mean the product of the elements of the triple. We need consider only primitive ...
real analysis	11/11/2007
Q: what is/are the proofs that rational numbers are a field? A: First we have to have an additive group. The rationals form an additive group because: 1. addition ...
Triplet	11/10/2007
Q: Sir, I am Mahima and I have 2 questions in this time. Please answer . 1. Is there a formula to ... A: This is a standard question, and there is a proof in almost all number theory text books. There are ...
Squarefree Number Theory	11/9/2007
Q: If b^2 is the largest square divisor of n, then a^2\|n implies a\|b A: Using the facts from you previous question, n = b^2*q where q is a prime or product of primes or ...
Squarefree Number Theory	11/9/2007
Q: An integer is called squarefree if it is not divisible by the square of any prime. Prove that for ... A: Consider the unique prime factorisation of n. For each prime whose index is even, e.g. p^(2k), then ...
Triplet	11/9/2007
Q: Sir, I am Mahima and I have 2 questions in this time. Please answer . 1. Is there a formula to ... A: 1. Yes there is a formula for all Pythagorean Triples. A primitive triple is one where there is no ...
Disibility by 9 and 11	11/7/2007
Q: First the question is working on mod 9 and mod 11. 2x99561=[3(523+x)]^2 On the left hand side I ... A: I'm not sure what you did on the right hand side. It is 3^2*(523+x)^2. The 3^2 = 0 mod 9, so we ...
Cantor's diagonal method	11/3/2007
Q: can you please explain Cantor's diagonal method? thanks A: There are two different interpretaions of this. The countability of the rationals and the ...
Simple Solution	11/3/2007
Q: I have a question that I believe you can answer. The answer should be determined without using any ... A: Yes, I think the following method could be done completely mentally. The LCM of 8,10 and 12 is 120. ...
Properties of Congruence	11/1/2007
Q: For n>= 1(n is greater or equal to 1) use congruence theory to establish: 27\|2^5n+1 + 5^5n+2. I ... A: I am afraid there is a misprint somewhere. it doesn't work when n=1. The expression evaluates to ...
Halloween Question	10/22/2007
Q: 8 quarts of cider were left in the punch bowl after the Halloween party. If, in addition to the ... A: One way to do this would be the following: we start with 8,0,0 fill the 3q and empty into the 5q We ...
statistics	10/21/2007
Q: A set of seven different positive integers has mean and median both equal to 20. What is the largest ... A: To make one of the values as large as possible, we make the 3 numbers smaller than 20 as small as ...
Number Theory	10/20/2007
Q: I have some questions on number theory. Can you help? 1) For which positive integers,n , is the sum ... A: In fact, we only need to consider one of the primes, since in multiplications and divisions with ...
Number Theory	10/16/2007
Q: I have some questions on number theory. Can you help? 1) For which positive integers,n , is the sum ... A: 1) The sum is n(n+1)/2 , and this is to be a factor of n! If n+1 is prime, this will not be a factor ...
Divisibility	10/16/2007
Q: a) Do there exist natural numbers m,n such that 7/17 = 1/m + 1/n ? b) Let p be a prime number. For ... A: a) The answer here is no. Clearing the fractions 7mn = 17(m+n) So either or both of m,n is a ...
Trig Identities	10/14/2007
Q: not sure if this question fits in the number theory category, but last year my maths teacher at ... A: It's not Number theory, but I promised to answer any Math question from a McAuley pupil. There's not ...
DIVISIBILITY THEOREM	10/11/2007
Q: Respected Sir, I would like to know about the main theorem of divisibility by 3,, as in 3 is ... A: This is based on what is called modulo arithmetic. Here the modulus is 3. Two numbers are congruent ...
Nth Term	10/9/2007
Q: theFind the rule for the n-th figure. Then find the number of tiles in the 200th figure. the 1st ... A: Did you notice that each diagram has 4 more tiles than the previous one. A sequence that you know ...
number theory	10/9/2007
Q: please see the following. 1. If p is an odd prime, let q = (p-1)/2 prove that (q!)^2 + (-1)^q is ... A: Now you are testing me. You didn't even tell me what the expression was congruent to. But I know ...
diophantine eqts.	10/8/2007
Q: Has anything been done on the nth degree pellian eqt. : X^n - AY^n = 1 where A>1, X,Y,n,A are in N ... A: I'm not sure that much has. It took long enough to sort n=2 completely satisfactorily. I did find a ...
congurance	10/8/2007
Q: please see the problems on congurences. 1) Let f be an integer-valued arithmetical function such ... A: 1) You haven't asked a question here. 2) The number n! is divisible by 2, but not by any other ...
Decimals & Fractions	10/1/2007
Q: Can you explain why or how the fraction 1/3 in 0.25 1/3 occupies the 1/100 place along with the 5 ... A: I am a number theory expert, not a business maths expert. Business people have their own funny ...
Decimals & Fractions	10/1/2007
Q: Can you explain why or how the fraction 1/3 in 0.25 1/3 occupies the 1/100 place along with the 5 ... A: I think the confusion arises because the notation you have used is not an accepted one. All the ...
Decimals & Fractions	10/1/2007
Q: Can you explain why or how the fraction 1/3 in 0.25 1/3 occupies the 1/100 place along with the 5 ... A: I'm not sure I understand the question. Do you want to add 0.25 + 1/3. If so, you need to convert ...
square	9/29/2007
Q: Q: If n is a natural number less than 100 and k is whole number ,for how many values of n, n^2=24k+1 ... A: I am always pleased when someone comes back with a follow-up. It means they are really thinking ...
square	9/28/2007
Q: Q: If n is a natural number less than 100 and k is whole number ,for how many values of n, n^2=24k+1 ... A: You certainly have a good starting point. As long as n is odd, the square is of the form 8m+1, so ...
number theory	9/27/2007
Q: Sir, Please see the following. Now I am asking not more than 3. pleaseeeeeeeeeee answer sir. 1. ... A: You do right to ask for clarification. WOLOG is a quick way of saying 'Without loss of generality'. ...
Fibonacci sequence	9/27/2007
Q: For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the ... A: If the 6th and 7th are a and b, then b^2-a^2 = 517= 1147 = 1517 Using the factors (b-a)(b+a) one ...
number theory	9/27/2007
Q: Sir, Please see the following. Now I am asking not more than 3. pleaseeeeeeeeeee answer sir. 1. ... A: 1) This is much more difficult to show than to prove there are an infinite number of primes of the ...
number theory	9/26/2007
Q: I am mahima asking again with following problems. please solve and show me. (1) Show that there are ... A: You are getting on to quite difficult work. 1) n!-1 is odd, so is divisible by at least one odd ...
congureance	9/26/2007
Q: Again I had the following please do it. 1)Show that if f(x)is a polynomial with integral ... A: That's a lot of questions! 1) Each term in the first polynomial b*a^n is congruent to the ...
congurence	9/26/2007
Q: prameela here. please see the following and give answer <1> Let a,b,m,n belongs to Z with m>0 and ... A: 1) Write m=nk a=b mod m implies a-b = tm =(tk)n implying a-b mod n. 2) a-b = tm, a-b = sn. ...
Number theory	9/25/2007
Q: What is the sequence of these numbers? 7,17,34,55 A: With so few terms, many sequences could start like this. One that does work (and the simplest ...
congurence	9/23/2007
Q: If (a, m) = 1 then there is an x such that ax is congruent to 1 (mod m). Any two such x are ... A: Helo mahima Elements 1,a,a^2,... form a group under multiplication mod m. Let a^r be the smallest ...
Number Theory	9/23/2007
Q: I'm having trouble getting started on a few problems prove/disprove a^2\|b^3, then a\|b. a^2\|b^2, ... A: All of these are proved by considering the unique prime factorisation of a and b. Let p be any prime ...
primes...	9/20/2007
Q: I am pavani. see the following! 1. Prove that 2 + square root of (-6) and 2 - square root of (-6) ... A: You are certainly making progress! These are complex numbers, and an important property of a complex ...
numbers and theory	9/20/2007
Q: Navya here. Please see the following question and solve it. (1) What is the largest number of ... A: 1) Every 4th integer is a multiple of 4, so you can't have more than 3 consecutive square free ...
Functions	9/20/2007
Q: Two functions are defined as follows: F(x) = x2+8x-18 and G(x) = 3x-6-x2. IF H(x) = ... A: You need to find where the functions have equal values. That is at x=-4 F(x) = G(x) = -34 and ...
merrsenne	9/20/2007
Q: I had the following doubt. In my tet book the following theorem given but proof not given in the ... A: I realise that I didn't read the question carefully enough. I proved that Mp was of the form 2kp+1. ...
need help with homework	9/20/2007
Q: Do you know anything about In-Out table. I'm a 9th grader in Highschool. Have you ever hard of ... A: I'd never heard of this before, but searching the web suggests to me that it is just a table of ...
fibnoci and lucas	9/19/2007
Q: Sir, In the Binet’s used in his Formula, phi^2 + phi =1. But Phi = 1.6180339887499… When we put ... A: You don't have the correct formula. It is phi^2 - phi =1. 1) This can be proved by induction. It ...
Fibbonacci Numbers	9/18/2007
Q: Please explain the following. n Find a formula for sigma Fi. ... A: To do this one you need to know the formula for F(n). It is (phi^n - (1-phi)^n)/5^(1/2) where phi = ...
divisions	9/17/2007
Q: I am prameela. I am doing number theory by your motivation and help in this subject. I cannot forget ... A: You are making me work very hard asking multiple questions. Please don't ask more than two ...
number theory	9/17/2007
Q: I came with new/diffrent problems. Please go throgh this and solve for me... 1) Prove that there are ... A: I hope you have been having a go at doing these for yourself. 1)Recall that a\|b implies a\|a+b and ...
plez	9/17/2007
Q: you never given help in my studies. I am so sad in this part. As an expert why you are not showing ... A: Let a and b be two distinct elements in S. Then a\|b and b\|a. This means a=b, a contradiction. So ...
n-theory	9/17/2007
Q: Sir, I am navya, I had the following 2 problems, which are eating my brain. please help me. 1. Let ... A: There is something strange about 1. What do the commas mean? Should there be P(n+1)? 2. Write ...
number theory	9/17/2007
Q: I came again with 3 problems. (1) Prove that every positive integer n, there are n consecutive ... A: 1) The n integers (n+1)!+r as r goes from 2 to n+1 all have the factor r. e.g. n=4. 5!=120. The ...
numbers	9/14/2007
Q: What is the sum of all 7 digit numbers using four 1's and 3 zeroes Its getting lenthier .... A: The number of arrangements of n things when p are alike of one kind and q alike of another kind is ...
number theory problems	9/12/2007
Q: Can you help me with these problems please. Find the value of: Show that every positive integer can ... A: This was quite a difficult problem. We choose the largest prime p that is less than or equal to n. ...
Decimal Fractional Numbers	9/12/2007
Q: Why do mathematicians dislike decimal fractional numbers? Here is an example: 0.66666..... is ... A: If a mathematician (especially an applied mathematician) is only interested in a final numerical ...
number theory problems	9/12/2007
Q: Can you help me with these problems please. Find the value of: Show that every positive integer can ... A: This is simply writing the number in base 2. We can just write down the highest power of 2 less ...
series	9/11/2007
Q: How R U? Sir,please tell me your date of birth... See the my new doubts and solve for me. A) The ... A: I can only make a conjecture at present. I suspect that if a is odd the gcd is 2, and if a is even ...
Number theory	9/10/2007
Q: I am navya from india. I am studying number theory with your inspiration. while studying this I got ... A: I help people to understand Mathematics for free for a number of reasons. Firstly I am a committed ...
series	9/10/2007
Q: How R U? Sir,please tell me your date of birth... See the my new doubts and solve for me. A) The ... A: There is still a lack of brackets here. a^2^m can mean (a^2)^m or a^(2^m). These are not the same. ...
series	9/10/2007
Q: How R U? Sir,please tell me your date of birth... See the my new doubts and solve for me. A) The ... A: A) This simply depends on the fact that (a+b,a)=(a,b) and (A1,A2)=1. You could set out a formal ...
Number theory	9/10/2007
Q: I am navya from india. I am studying number theory with your inspiration. while studying this I got ... A: 1) If n is even then it is the sum of 4 and n-4, both even and therefore composite. If n is odd, it ...
divisions	9/10/2007
Q: Sir, I am Mahima. Please see the following. 1) For each of the following give a proof with counter ... A: (a) Write (a,b)=h and a=sh,b=th then (s,t)=1 b^n/a^n=t^nh^n/(s^nh^n)=t^n/s^n is an integer. ...
nth terms in sequences	9/9/2007
Q: it says generate the first 10 terms of these 1)n squared + 5 2)2n squared + 5 3)n squared + 10 4)4n ... A: It is just a matter of plugging in the numbers 1 to 10 for n. I will work out the first three of ...
Sexagesimal	9/9/2007
Q: How would I use the Sexagesimal system? Unlike the decimal system where each step up is a multiple ... A: Sexagesimal is base 60, but we don't use 60 different symbols for the digits. Instead base 10 is ...
number theory	9/8/2007
Q: please see the following and give me a solution 1) For integers a,b with b < 0, show that b < a mod ... A: I asked you to try an example of these. This is the only way you are going to make progress. Take ...
doubts	9/7/2007
Q: Sir, What do you mean by conjecture? Is it theorem or ... Also, I have very big doubt in my mind. ... A: A conjecture is simply a guess. In other words a theorem that seems to be correct but for which ...
solution of problem with venn diagram	9/6/2007
Q: Out of 200 students, 50 of them takes discreate mathematics, 140 of them takes economics and rest ... A: Your question is still not making sense. If 30 take both DM and P, how can only 10 take P? In your ...
sum of squsres	9/6/2007
Q: In how many ways a number can be written as the sum of two sauares? for example: 5 = 1^2 + 2^2; ... A: There is a simple formula for the number of ways. First factorise the number into the form ...
solution of problem with venn diagram	9/5/2007
Q: Out of 200 students, 50 of them takes discreate mathematics, 140 of them takes economics and rest ... A: The question is not clear. It mentions both courses, but there seem to be 3 different courses. ...
numbers	9/3/2007
Q: Imagine that the digit 7 has been banned from use for superstitious reasons. So what would call "the ... A: Since one digit is missed out, this notation must be very similar to base 9. Checking easily shows ...
problem about counting	9/3/2007
Q: How many terms are contained in (a+b+c)^20? A: I have to assume you mean different terms. If the index of a = ind(a) is 0, then ind(b) has 21 ...
number theory	9/2/2007
Q: good evening. Thank you for answering the previous questions. Please answer the following problems. ... A: All of these are self-evident and do not need proof. Z is the set of integers, so that a is an ...
induction	9/2/2007
Q: I am pavani. I had the doubt(s) in induction. please clarify with examples. 1. What do you mean by ... A: 1. The weak version is: Given a theorem T(n), T(1) true and T(k) true implies T(k+1) true, then ...
numbers	8/31/2007
Q: How many integers between 1 and 1000000 have the sum of digits equal to 18? What is the generalzed ... A: I've just found this in the question pool. The integers all must be multiples of 9, but that doesn't ...
theory of numbers	8/30/2007
Q: sir, I came again to know the folowing problems. ax^2 + 2bxy + cy^2 is a polynomial. when a, 2b, c ... A: Consider any one prime p which is a factor of both a and b. a = ...p^x.... and b = ....p^y... in ...
Proofs	8/29/2007
Q: Can you please help me withe following: Is it True or False and prove it: 1)A+BC=(A+B)(A+C) i ... A: This isn't number theory, it is Boolean Algebra, the algebra of logic. The symbols represent the ...
6 digit combo	8/29/2007
Q: What is the total number of combinations on a lock?? (digits 0-9) A: Note that although the usual phrase is 'combination lock', it is actually a permutation lock since ...
divisions.	8/28/2007
Q: I did not undrestand these questions. please help me. 1. if (a,b) =1 then (a^n,b^k) =1 for all n>=1, ... A: 1. The only factors of a^n are a,a^2,a^3...a^n The only factors of b^k are b,b^2,b^3...b^k. These ...
theory of numbers	8/28/2007
Q: sir, I came again to know the folowing problems. ax^2 + 2bxy + cy^2 is a polynomial. when a, 2b, c ... A: 1. It is not possible to find a polynomial that gives only prime values. 2. This is obvious if ...
number theory area	8/27/2007
Q: I am Mahima came again with 3 questions. please solve and encorage me in my studies. 1. how to prove ... A: These questions are basic book knowledge which can be found in any number theory book or a tutorial ...
do it	8/27/2007
Q: i am navya.Does there always exist atleast one prime between n^2 and n^2 + n for every integer n>1? ... A: This is an unsolved problem of number theory. In fact, also unsolved is the weaker bound. There ...
number theory	8/26/2007
Q: what is the algebraic solution for x, y, z to the following eqn : x^3 + y^3 = z^3 + 1 A: I don't know one, and I haven't been able to find one. However, if you go to ...
Ages	8/24/2007
Q: Hey Raja, this is more of a riddle than a math question however I think you will be smart enough to ... A: I have had another look at the problem, and as long as you have typed the riddle exactly as in the ...
Probability	8/24/2007
Q: I'm very grateful for the solutions to my previous questions.Your approach to the problems have been ... A: DPAB is a re-entrant quadrilateral like an arrow head, so the probability is zero. If this was a ...
Base	8/23/2007
Q: Find, as a proper base ten fraction, the sum of the following infinite series of repeating decimals, ... A: Perhaps it is easier to consider 0.999999999.... You would have to agree that this is the same as ...
Remainder	8/23/2007
Q: Find the last 3 digits of 7^347 I know we can get ans by dividing the no by 1000,but no theory is ... A: 7^4 = 1 mod 5 by Fermat's theorem. In fact 7^4 = 2401 =1 mod 100. We need to find the smallest ...
Base	8/23/2007
Q: Find, as a proper base ten fraction, the sum of the following infinite series of repeating decimals, ... A: This is a lot easier than you imagined. Consider the number .0999... in base 10. It is just another ...
Rates	8/21/2007
Q: If ten workers perform one job in 5 days, one person performs one job in how many days? Here is ... A: You seem to be multiplying workers X jobs X days. This not the method. We multiply workers by days ...
numbers	8/20/2007
Q: please solve the folowing ones., 1) For integers a, b with b <0, show that b<a mod b < or = 0. 2) If ... A: 1. I have never seen a mod b with b<0. However, a mod b is just the same as a mod -b. The problem ...
general	8/20/2007
Q: sir, I am MAHIMA from india. I had the following questions. please answer thease as early as ... A: 1. If n is a square, its square root is the number required. If not it factorises into two factors ...
number theory	8/15/2007
Q: I cam back. How are you? I am fine. I hope you are also fine. I had the following quarries. please ... A: I will try to explain the meaning of each of these terms by means of an example. A theorem is a ...
Probability	8/15/2007
Q: A circle contains 2006 points chosen so that the arcs between any two adjacent points are equal. ... A: The total number of triangles is 2006C3. For a right angled triangle, the hypotenuse is a diameter. ...
raising 2005 to the power 2005	8/13/2007
Q: When written as a decimal number, 2005^2005 has D digits and leading digit L. Find D + L. A: We start by taking the log to base 10 of the number and using the laws of logs. This gives ...
how to solve these type of problems	8/7/2007
Q: good evening sir, I am one of the very intrested one who want to learn number theory properly. I had ... A: Every child in junior school knows the division algorithm but almost certainly doesn't call it that. ...
proove it	8/7/2007
Q: Reapected sir, I came again to get help from you. How r U sir? why u r not available from last 1 ... A: I was on holiday last week with no internet access. 1. You have probably used (a,b) = (a,b-a). This ...
Numbers	8/6/2007
Q: How many integer solution exists for \|x\|+\|y\|+\|z\| = 15 Pls help on this..how do i proceed..taking 1 ... A: I have just found this in the question pool. The best way is to think geometrically. The points are ...
Ages	8/6/2007
Q: Hey Raja, this is more of a riddle than a math question however I think you will be smart enough to ... A: Your answer is not correct because Jim is 36 and that is 3 times as old as your value for Sid's age ...
Numbers	7/27/2007
Q: How many natural numbers having at most six digits have the sum of their digits as at most six? A: We first choose a selection of 6 digits e.g. 122000 and then arrange them. A selection of n objects ...
twins and sophie germain primes	7/24/2007
Q: sir, so far do we know of any relation between twin primes and sophie germain primes? given the ... A: It has not been proved that there are an infinite number of either of these. Attempts have been made ...
primes in algebraic series	7/22/2007
Q: sir, just like we have twin primes (p, p+2) cousin primes (p, p+4) sexy primes (p,p+6) sophie ... A: I am sorry to say, very little. I am not aware of any work done on this subject. It has not even ...
integer sequences	7/17/2007
Q: My question concerns the numbers produced by two different sequences. I am wondering if it is ... A: This is a fascinating problem. I assumed from the example that a and b had to be positive integers. ...
integer sequences	7/16/2007
Q: My question concerns the numbers produced by two different sequences. I am wondering if it is ... A: No-one has asked me one like this before. Such an equation is called a Diophantine equation after ...
maths subject	7/16/2007
Q: I am trying to find out the name of a particular mathematics subject. But I do not know what it is ... A: I don't think there is a name for the topic, but it comes up in most algebra courses. You may find ...
Unique numbers	7/15/2007
Q: I have a list A of unique numbers of 9 digit each. I generate another list B by interchanging right ... A: First, about the 1:1 transformation. Of course, squaring is not 1:1 if the domain is the set of ...
Rates	7/14/2007
Q: Here is myIf 15 workers can perform a piece of work in 7 days, each worker performs what fractional ... A: We have to assume here that each worker does the same amount of work, and that each does the same ...
Unique numbers	7/14/2007
Q: I have a list A of unique numbers of 9 digit each. I generate another list B by interchanging right ... A: We must assume that the interchange retains the order of the two blocks of digits. If the left 4 ...
Application of number theory	7/9/2007
Q: Do you know what are applications of number theory other than cryptography? Thank you, Frans ... A: This must be a very common problem. If you can apply theorems when you have worked examples at hand ...
Application of number theory	7/8/2007
Q: Do you know what are applications of number theory other than cryptography? Thank you, Frans A: I don't know of many, but I have chosen the Chinese Remainder theorem as an example. It was ...
number theory pattern question	6/8/2007
Q: whatsw the next three terms of the pattern: 4,6,9,10,14,15,21,22,25,26,33,34,35,38 i am thinking the ... A: Well done! These numbers are called semiprimes. Of course, since there is no formula for the nth ...
number theory	6/4/2007
Q: how r u? I hope you are fine. according to your valuble suggextions, I am studying number theory ... A: A Mersenne prime is a prime of the form 2^n - 1. If n=3, we have the mersenne prime 7. To each ...
Box Principle	5/23/2007
Q: Good Evening! I'm having a problem with these two numbers, basically I have an idea but I don't ... A: I can now help you with your third question The subset must contain one at least of the pairs 1,2 ...
math problem	5/15/2007
Q: the diagram is one of a graph of the function f(x)=(x^2)+1,-1<=x<=1 and p,q E(element) R.(i)find ... A: This problem was structured to get you to understand when a function had an inverse. Unfortunately, ...
Interest Rates	5/14/2007
Q: The amount of $1.00 compounded annually at 4% for 5 years (5 periods) is $1.216653. For a principal ... A: Tables do not normally have units in them because it is wasteful of space to put the same unit in ...
n-theory	5/8/2007
Q: sir, I would like to learn numbertheory from elementary to advanced level. please tell me the book ... A: I think you can use google or another search engine to find other sites yourself. If you have worked ...
Number combinations	5/7/2007
Q: How many different combinations of 13579 can be generated and then rank them from lowest to highest. ... A: I think there is a problem with the language used here. The word combination refers to a selection, ...
n-theory	5/6/2007
Q: sir, I would like to learn numbertheory from elementary to advanced level. please tell me the book ... A: I have just seen an inexpensive book Elementary Theory of Numbers (Dover Books on Advanced ...
Fermat Difference of Squares	4/27/2007
Q: I have noticed that everyone solves Fermat's Difference of Squares by taking the square root of the ... A: Excellent. I have just done the analysis myself. (2m-1)^2 - (2m)^2 gives a number of the form 4k+1, ...
Fermat Difference of Squares	4/27/2007
Q: I have noticed that everyone solves Fermat's Difference of Squares by taking the square root of the ... A: I am afraid that there is no way to know whether the number to be tested for primality can be ...
need help	4/25/2007
Q: The Egyptians used the square of 8/9 of the diameter (=(8d/9)2) for the area of a circle. In ... A: Substituting for d =2r and squaring gives k = 256/81 This is approx 3.160493827 compared with PI ...
math	4/24/2007
Q: Vijilant, Please help. I do not know how to solve 2 = 14x + 60y using Euclidean's algorithm. Wendy A: The Euclidean algorithm, or Euclid's algorithm is the method for finding the gcd of two or more ...
powers and common multiples and or divisibility	4/11/2007
Q: my horrendously vague stab in the dark at a title in the subject line should take care of clarifying ... A: 1. No, this isn't true. Just consider 10 and 8. 10 = 2*5 and 8 = 2^3. 10 contains the factor 5, ...
powers and common multiples and or divisibility	4/10/2007
Q: my horrendously vague stab in the dark at a title in the subject line should take care of clarifying ... A: This is an easy one to explain. There are no powers of two that are multiples of 10. This is ...
probability of a specific digit sequence within irrational numbers	4/7/2007
Q: the probability that a finite sequence of digit like 1618033988749 is contained within a specific ... A: I am pretty sure that this has never been proved. There are quite a few web-sites with interesting ...
prime pattern	4/4/2007
Q: vijilan. the question is: The smallest positive integer n for which the numbers n^2+1, n^2+3, n^2+7, ... A: I see what you mean by brute force. I can certainly improve the time very quickly by a factor of ...
prime pattern	4/3/2007
Q: vijilan. the question is: The smallest positive integer n for which the numbers n^2+1, n^2+3, n^2+7, ... A: I have thought about this again and have come to the conclusion that quadratic reciprocity may not ...
prime pattern	3/30/2007
Q: vijilan. the question is: The smallest positive integer n for which the numbers n^2+1, n^2+3, n^2+7, ... A: This is most definitely a research question and one which I can't answer right away. You haven't ...
number theory	3/28/2007
Q: Generate a random 200-digit number, what is the chance of getting a prime number? A: I am afraid I do not have the computing power to calculate this exactly, but I can estimate it using ...
curious relationship	3/19/2007
Q: playing with some numbers I noticed a pattern and I was wondering what this was called and where I ... A: The process of summing the digits of a number repetitively until a single digit remains is called ...
puzzle	3/13/2007
Q: you did not answer. The word is yo yo and each direction counts. What is the formula?i have a word ... A: I will have to assume that the arrangement is: YOYOY OYOYO YOYOY OYOYO YOYOY There is one YOYO in ...
number theory	3/5/2007
Q: 1. Find all positive numbers x and y which solve the equation 41x 16y = 2000. 2. Find the ... A: The first one has a sign missing. I shall assume it is + Since 16 is a factor of 2000, write x=16X ...
math	3/4/2007
Q: that's what i want the program for is to check the primes and if its programmable to save a file of ... A: I have found a book which contains and explains algorithms for testing and finding prime numbers. ...
quadractic equations	2/28/2007
Q: Please help me solve these..if you can. choose a value for k so that the equation 5x^2-10x+7=k ... A: This is an exercise in completing the square. You should be able to complete the square on the LHS ...
math	2/26/2007
Q: wrong e mail address I am looking for a program that can do math in large numbers over 10 million ... A: Sorry for the delay. I forgot to log my holiday. Mathematica can certainly deal with numbers with ...
Fibonacci series	2/1/2007
Q: I have a conjecture if f(n) is the nth term of the fibonacci series, f(n-1) + f(n+1) -1 = 0 mod n ... A: I am delighted to tell you that I have now constructed a proof of your conjecture. L(n) = ...
combinitorics question	1/25/2007
Q: a class of 10 boys and 15 girls are divided into five groups. how many different ways can the ... A: I don't claim that I am an expert on this topic but I can help you with your question. I assume from ...
expectation	1/19/2007
Q: If you want to find the variance of X: let S denote each traingle in X, then you observe that Var(S) ... A: Apart from the mistype of = instead of -, I agree with your calculation of the variance of x in the ...
expectation	1/18/2007
Q: One of my friends told me about your problem-solving skills. I am trying to solve this problem, ... A: I see no reason to doubt your first analysis. My first thought always in a tricky situation is to ...
combinations from limited duplicative sets	1/10/2007
Q: I'm embarrassed to discover that I sent you a typographical error in the problem that I posed. The ... A: Yes, I thought it may have been a mistype. Now, (p+1)^2 has 21 factors, so must be of the form ...
combinations from limited duplicative sets	1/9/2007
Q: Vijilant, I don't have a folow up question, but I just wanted to thank you for your answer to my ... A: The number of factors of a number is an example of what is called in number theory, a multiplicative ...
combinations from limited duplicative sets	1/8/2007
Q: Suppose you have n objects, but only m are distinct. That is, there are n1 objects of the 1st kind, ... A: This is a combinatorics question which is not my field of expertise. I don't know a formula for ...
0.9r	12/30/2006
Q: ah thanks that clears up my last query, so if the limit of a recurring number is the number which ... A: This will have to be the last post on this topic. The limit of the series 0.9 + 0.09 + 0.009 +.... ...
0.9r	12/29/2006
Q: so are you saying that some mathematicians accept and use limits and others dont, or have I ... A: Mathematicians everywhere accept and use limits. What some disagree about is whether it is possible ...
0.9r	12/29/2006
Q: is the limit of a number just an idea that someone made up or is it actually something, lets say ... A: You are talking like a real mathematician:- being sceptical and not accepting anything without ...
Cooking Measurements	12/25/2006
Q: Can you please tell me what 2 ounces are equaled to? A: If you are talking about weight, 1 ounce = 28.3495 gm. Since this is not a nice round number, if you ...
primes	12/22/2006
Q: sir, What is the diffrence between reletive prime, twinprime and co prime. thanks. A: Two numbers which are relatively prime are numbers whose gcd is 1. For example 6 and 25. Neither ...
number theory	12/17/2006
Q: need to know if i am going right.Prove by mathematical induction, that if a,b elements N then ... A: I've never used induction to prove this because (a^n+b^n)/(a+b) is the sum of a GP when n is odd. ...
congruency	12/16/2006
Q: prove that 111333+333111 is divisible by 7. A: I am sorry to disappoint you. Perhaps you have mistyped the expression. 111 = -1 mod 7, so 333 = -3 ...
numbers/euclid	12/12/2006
Q: Sir, I am Pavani. I don't know how to solve the following problems, please answer for me. 1. How to ... A: The Euclidean algorithm finds the gcd of two numbers by repeatedly replacing the larger with the ...
Numbers with exactly 7 prime factors	12/11/2006
Q: Another similar question is : What is the smallest set of consecutive 3 numbers with 6,7 & 8 prime ... A: Once again, this does not seem to be a problem that number theory can shed much light on, except to ...
primes	12/8/2006
Q: I am your student pavani. how are you? I got doubt in the folloing poblem. If the number is small, ... A: I am afraid there is no easy method to factor a product of large primes. Fermat used a trial method ...
Average Number of Prime Factors	12/6/2006
Q: It seems to me that as an integer n increases, the chances increase that it will have many prime ... A: You are quite right in your conjecture that the chances increase as n increases. A few formulae are ...
Numbers with exactly 7 prime factors	12/5/2006
Q: Is there a rule to determine all numbers that have exactly 7 prime factors ( say less than 1 ... A: When a number is written in Euclidean prime factor form as in your examples it is simple to add the ...
theory of congruences	12/3/2006
Q: how can we verify that 0,1,2,2^2,2^3,.......,2^9 form a complete set of residues modulo 11.but that ... A: I don't know if you have any experience of elementary group theory, but if you do this is the ...
theory of congruences	12/2/2006
Q: what is remainder when 41^65 is divided by 7 A: I can't type the congruence symbol, so will type = instead. 41 = -1 mod 7 41^2 =(-1)^2 = 1 mod 7 ...
combinations	12/1/2006
Q: How many possible 6-digit number combinations can you make using the digits one through NINE? A: Assuming you are not allowed to repeat any digit and you are not interested in the order of the ...
theory of congruences	11/30/2006
Q: prove that the integer 53 raised to 103 +103 raised to 53 is divisible by 39 A: The method is to prove it is congruent to zero mod 3 and also mod 13. 53 = 2 mod 3 53^2 = 4 = 1 mod ...
reply2 : Circles	11/24/2006
Q: I downloaded the trial version of mathematica. It had the minimize function. I don't think i'm ... A: Perhaps your syntax wasn't quite right. I typed ...
reply : Circles	11/23/2006
Q: Yay, your the only expert that actually thought about my question :) I understand the bit about the ... A: I loved that graphic. It's similar to the one I tried with winplot but better. I had a look in ...
Circles	11/23/2006
Q: i couldn't find anyone else that could answer this: "this question has been denied by other ... A: This is certainly not a problem suitable for calculus because of the abs and sqr functions. Note ...
Basic functions	11/23/2006
Q: I don't know exactly what number theory is exactly, but I figured you might best be able to answer ... A: I gave you the example of the Euler function which is relatively simple to explain; there are also ...
Basic functions	11/21/2006
Q: I don't know exactly what number theory is exactly, but I figured you might best be able to answer ... A: Sorry for the delay in answering your question. I have been very busy marking exams. There are many ...
Quadratic Residues	11/9/2006
Q: It makes perfect sense. Your argument can also be used to show that 3.2^(n-1))^3 is always a ... A: This is a bit easier than I thought it was going to be. The group is of order 12nk+6 =3(4nk+2) So ...
Combinations of 3	11/8/2006
Q: If I have 12 different items. How many different combinations of 3 items can I have? i.e. 12 ... A: The mathematical notation for this is 12C3. That is 12 choose 3. It is calculated as ...
Quadratic Residues	11/3/2006
Q: I have strong evidence that (3.2^(n-1))^3 is always a quadratic residue of primes of the form ... A: I assume you are familiar with the quadratic reciprocity law, the legendre symbol, which I shall ...
maths	10/30/2006
Q: what is the nth term of this number pattern 0 7 16 27 40 A: There are a number of different ways to look at the problem. The obvious approach is to look at the ...
number puzzle	10/18/2006
Q: add numbers wheels on eight taxis even numbers up to ten numbers on one dice A: This is a question which is capable of several different interpretations, for instance does it mean ...
Big O Notation	10/17/2006
Q: Can you help understanding the Big O notation. I have difficulties in determining the O notation of ... A: This is not really my expertise. The only time I remember using the notation in number theory was ...
Divisibility Proof	10/14/2006
Q: I cannot seem to figure out the proof by contraposition for the following: For all integers a,b,and ... A: This is an interesting question, because it seeks to see if you understand the different types of ...
base 10	10/13/2006
Q: Why do we count on base 10? What are the advantages (other than having 10 fingers - I've never seen ... A: Historically, Im pretty sure you are right in thinking that it was a good system because for example ...
Whole Numbers	10/9/2006
Q: What is the least number that leaves a remainder of 3 when divided by 5, a remainder of 2 when ... A: You didn't tell me what your mathematical background was, so I assumed you had met the modulus ...
Proof	9/27/2006
Q: prove that for any m,n 'E'(element) N(nat.nos.) m not= n --> m^2+n^2>n^2 i tried this...(since m ... A: I think you may have mistyped the question. Since m^2>0 for n a natural number, m^2+n^2>n^2, so it ...
Discrete Math	9/21/2006
Q: Your guidance was pivotal in getting a B on my last test and now I have another study guide I am ... A: Congrats on your B. I can't see how the hint helps. I looked at the ratio n!/40^n and what happens ...
MAth	9/17/2006
Q: which number completes the sequence 2, 9, 5, 13, 10, 19, 17 Answer a. 22 b. 24 c. 25 d. 27 e. ... A: If we look at the pattern of differences; that is the difference, term - previous term we have: ...
Math	9/17/2006
Q: someone asks you for a sum of money between 1 dollar and 5 dollars. I'm going to give you five ... A: Instead of using base 1 we use base 2. In the envelopes we put 1,2,4,8,16,32,64,128,256 and 512 ...
number theory	9/12/2006
Q: two questions: 1. what is a sucessive product of a number called? 2. what is a name for a product ... A: There could be a number of different interpretations of both your questions. I am assuming that ...
Number theory	9/11/2006
Q: Find the gcd (243,1025)unsing the euclidean algorithm, and express the gcd in terms of the original ... A: The Euclidan algorithm proceeds by successively dividing the most recent divisor by the most recent ...
Days of the Week - Congruence	9/10/2006
Q: I was so happy and suprised to get such a quick response. When I worked on the problem, I also ... A: I covered the point about differences in my reply. The point is that there are 4 years in every 28 ...
set theory	9/9/2006
Q: there are 50 people ,20 like football and criket ,10 like tennis and criket ,35 like tennis and ... A: You have changed the question, but the answer is the same. There is not enough information. You ...
Days of the Week - Congruence	9/9/2006
Q: If you're familiar with the congruence formula to work out the day of the week, can you please help ... A: Looking at the congruence d=N+[2.6M-0.2]+ Y+[Y/4] +[C/4]-2C -(1+L)[M/11] mod 7 When we keep the ...
set theory	9/9/2006
Q: there are 50 people 20 like jukebox + games, 10 like games + book , and 35 like book + juke box ... A: There does not appear to be sufficient data here to answer the question. I assume that the 35 who ...
babys looks	9/6/2006
Q: i have brown hair and brown very brown eyes my father does also.my mother has blue eyes and blonde ... A: This is not really a question for a number theory expert, but I will give you an analysis based on ...
euclid's algorithm	9/5/2006
Q: ok, how did u get s(x) and t(x)? d(x)=-1/10(4x+11)a(x) = 1/10(4x^2-x-1)b(x) where does ... A: I have obviously overestimated your algebraic manipulation ability. I will fill in the gaps. From ...
euclid's algorithm	9/5/2006
Q: c(x)=8x^2-14x-4 The second division gives 32b(x)=(4x+11)c(x)+10d(x) I am sure you will be able to ... A: The point about the Euclidean algorithm is that it proceeds by successive divisions. To do an ...
euclid's algorithm	9/5/2006
Q: you say to multiply 32b. how do i know what amount to multiply by? could i have used any multiple so ... A: A good question Daniel! If we are trying to divide x^3+.... by 8x^2+.... it is obvious we are going ...
euclid's algorithm	9/5/2006
Q: find greatest common divisor of: a(x) = (x^4) -2(x^3)-(x)+2 b(x) = (x^3) + (x^2)-5(x)-2 express d(x) ... A: We are not interested in numerical factors here and we know that any gcd must be monic(leading ...
Equivalence Relations	9/4/2006
Q: I understand equivalence relations but this one on my study guide threw me for a loop. Please help! ... A: I am not sure if you know the definition of an equivalence relation. About 50 years ago I learned ...
Math Induction	9/4/2006
Q: On proofs like this is there a standard way to determine each step based on the setup or is each ... A: I am afraid that there is no easy answer. You just have to study different types. In the ...
Math Induction	9/4/2006
Q: You answered a question I had before and I appreciate it I have an example I am working to study for ... A: This is very similar to the previous one, though you have missed out some brackets. It should read ...
Mathematical Proofs	9/4/2006
Q: I am hoping I am in the right area to ask for help. I need help in setting up and proving the ... A: Yes, you are in the right area here. I hope that you are familiar with proofs by induction because ...
Unity symbol	8/16/2006
Q: I am looking for a unity symbol, perhaps taken from math symbols, to use in a graphic representing a ... A: The usual meaning for the capital U is for the union of two sets, for example AUB is the notation ...
Modular exponentiation	8/11/2006
Q: I am grateful that you gave an explaination to the RSA problem. But you did not give an answer to my ... A: Now I understand what you mean. Given the size of n and the fact that we dont know phi(n), I don`t ...
Modular exponentiation	8/10/2006
Q: I am trying to understand the subject of modular exponenation for RSA problem, i.e. x^y mod n = z. ... A: Let me just go through the process of RSA. Two large primes p and q are selected. n=pq. n forms ...
number puzzle	8/2/2006
Q: Its a question on the Mint. At the top it says add all the numbers, then it gives the three puzzles. ... A: I have already answered your question. If you have a calculator, try inputting 4+3X2. If it is a ...
number puzzle	8/2/2006
Q: a puzzle is set as follows. Nineteen minus five = 4+3x2 = 6-5= then says add all ... A: I can't believe this is the full text of the puzzle. For a start there isn't a question. Is the ...
chaos in number theory	7/16/2006
Q: sir, iam interested in knowing about the applicatons of chaos theory to number theory. in ... A: Matrix chaos mathematics has been used in the study of the Riemann Zeta function zeroes. See for ...
perfect equations	7/12/2006
Q: I am a designer and fine artist. I have also studied Philosophy at Birkbeck college in London. As ... A: I'm busy marking exams at present so don't have time to write a thesis on this subject. However, my ...
Riemann zeta function hypothesis	7/9/2006
Q: What wood be the best textbook to study this? Doing some research on it Thanks A: A lot depends on how much Mathematics you know. One that has had good reviews is Title: The Riemann ...
Inverses	7/8/2006
Q: If one U.S. dollar is equal to 115.68 Japanese yen, one Japanese yen is equal to how many U.S. ... A: Let's start with an example nearer home. 100 cents = 1 dollar What's 1 cent worth in dollars. We ...
theory numbers	7/4/2006
Q: We are a group of maths teachers following aproffessional course , we would be happy if you ... A: It is difficult to explain what the theory of numbers is about in a few short paragraphs. Most ...
Divisibility	7/1/2006
Q: How do i prove that the product of three consecutive numbers is divisible by three? A: If the lowest number is a multiple of 3, there is nothing to prove. If it is not, it (a) leaves ...
just the same!!!	6/30/2006
Q: ^^ it's me again, jaizelle...i thank you for your nice answer with my question but this time i'm ... A: I am afraid that there is no facility for including diagrams on this site, nor a facility for ...
set theory	6/29/2006
Q: sir it's me again, avon...what i mean is an example of a worded problem just the same as to my ... A: Here is another one. Perhaps you could make one up yourself after studying this one. In a class of ...
set thoery worded problem	6/28/2006
Q: 12 people in town have descided to form 3 clubs. Some are social than others, and the clubs are ... A: As it stands, your question doesn't make a lot of sense. I can't make out what 1/3 of the people ...
example of worded problemusing set theory	6/28/2006
Q: sir may i ask for a simple example of wored problem that applies the set theory...of course i need ... A: 100 students in a college belong to at least one of the Math, Physics and Chemistry societies. 50 ...
worded-problem applying set theory with set formulas	6/26/2006
Q: kindly give us an example of worded problem applying set theory with its set formulas and ... A: I cannot be sure what sort of question you want answered, but I will give an example which uses the ...
maths	6/19/2006
Q: numbers 3 3 8 8 can only be used once, but the symbols + - / x can be used as many times but the ... A: If you don't have to use all the numbers it's easy. Just 3X8. If you must use all the numbers, I'm ...
Complete the series etc.	6/14/2006
Q: Replace the ? with appropriate number- Q.1) 18, 14, 12, 11, ?, 10.25 Q.2) 5, 23, 113, 523, ? Q.3) ... A: Sorry I forgot to post that I would be away on holiday, and didn't get back till today. Q1. The ...
find the missing number p	6/3/2006
Q: I know that you have answered this question once but i was not able to get it. Could you answer it ... A: I have just realised that allexperts does not update previous answers as often as I thought and the ...
number theory:divisibility	5/27/2006
Q: The product of any three consecutive integers is divisible by 6. A: Exactly one of the 3 integers is divisible by 3. Either one or two of the integers is divisible by ...
fibonacci numbers	5/21/2006
Q: i have recently watch the new film the da vinci code. In this one of the first things that comes up ... A: Yes, fibonacci numbers are a fascinating field of study. I presume you know that they are the ...
Number combinations	5/21/2006
Q: I would like to know how many 5 digit number combinations you can do with 1 - 47 (i.e. 1 3 19 32 47 A: Sorry for the delay. I thought I had answered this question. Something must have gone wrong. The ...
Adding & Subtracting Fractions	5/19/2006
Q: Here is myA pole 22 5/6 feet long is broken in two. One piece is 2 3/4 feet longer than the other. ... A: Using the hint, we add 2 3/4 feet to the length of the shorter. The total length is then 22 5/6 + 2 ...
Number Theory	5/11/2006
Q: Given a number how do you find position in wythoff array ? A: The Zeckendorf expansion of a number is the sequence obtained by successively subtracting the ...
Complex Trigonometric Functions	4/14/2006
Q: http://mathworld.wolfram.com/Sine.html http://mathworld.wolfram.com/Cosine.html ... A: You seem to be confusing z, a complex number, with its argument. z can be written in cartesian form ...
An in-depth look at algebraic approximation of trignometric functions through infinite sums, and further results	4/14/2006
Q: I'm sure you're familiar with the Wallis product (if not, here's a link ... A: (a) You haven't given all the details. I would be interested to know how you differentiated your ...
An in-depth look at algebraic approximation of trignometric functions through infinite sums, and further results	4/14/2006
Q: I'm sure you're familiar with the Wallis product (if not, here's a link ... A: I don't fell qualified to recommend books on calculus or algebra, but you can find plenty of good ...
An in-depth look at algebraic approximation of trignometric functions through infinite sums, and further results	4/14/2006
Q: I'm sure you're familiar with the Wallis product (if not, here's a link ... A: First of all, please realise that calculus is not my area of expertise. I have checked your formula ...
Fractions	4/13/2006
Q: I am having some difficulty solving the following: 2/3 of A's money increased by 4/5 of A's money ... A: Your problem is really caused by the archaic language in the problem. The word 'lacks' is not used ...
Recommended Books	4/11/2006
Q: What is your opinion on recommended books about algebraic number theory and analytical number ... A: I think a lot depends on your level of expertise. If you are an undergraduate, 'A brief guide to ...
group theory	4/7/2006
Q: Let H, N, and M be subgroups of a group G. Verify the following. (a) If N is normal in G then HN = ... A: Another abstract algebra question. (a) N is normal implies hN = Nh. That is the easily remembered ...
Ring Theory	4/7/2006
Q: (a) Prove that the set S of rational numbers (in lowest term) with odd denominators is a subring of ... A: Once again this is not my main area of expertise. I have time to answer 2 questions and will try to ...
Introduction to group theory	4/6/2006
Q: Let G be a group of cardinality pq where p and q are distinct primes. Show \|Z(G)\| = 1 or pq Thank ... A: I thought I had answered this question, but the answer must have gone astray. I am going to answer ...
group theory	4/4/2006
Q: Let G be a group of cardinality pq, where p and q are distinct elements primes. Show \|Z(G)\|=1 or pq ... A: This is not really my area of expertise, but a number theorist has to know something about group ...
various infinities	3/27/2006
Q: I would like to have a definition of a "curve". Given a definition, I would like to ask "Is the ... A: In Cantorean theory a curve is another example of a continuum. Just imagine it as a piece of string ...
Fractions & Ratios	3/12/2006
Q: What simple solution, one not requiring algebra if possible, can be used to determine the missing ... A: It is possible for the first unknown to have any value whatsoever. This is because there are two ...
roman numbers	2/21/2006
Q: what is AD MDCCCXCV in real numbers A: The Roman numeral representation of a number is just as real as the decimal system we use. M=1000, ...
What are the advantages of...	2/12/2006
Q: What are the advantages of the hindu-arabic system(present decimal system)over the other ancient ... A: Western systems used physical representations for numbers. For instance, lines scribed in the sand ...
Number Theory-Primes	2/5/2006
Q: 1.) if p > or equal to q> or equal to 5 and p and q are both primes, proved that 24 \| p^2-q^2. 2.) ... A: It's great to find females studying number theory. A prime other than 2 or 3 can only be congruent ...
Abstract Algebra - cyclic groups	2/4/2006
Q: If you have two elements from a group whose orders are relatively prime to one another, how do you ... A: Suppose to the contrary. Then there is an element a which is in both subgroups. Consider the ...
fermats thm	2/4/2006
Q: will you please expl me in simple words the fermat's ittle theorem? mayur A: Fermat's "little" theorem states that if p is prime and a is prime to p, then a^(p-1) is congruent ...
numbers 0 1 2 3 4 5 6 7 8 9	2/2/2006
Q: please i would like to know 1 where are the decimal system came from, arabs, indians, others .....? ... A: I am not an expert on history of the number system, but here is a link to get you started. Clearly, ...
divisibility	2/1/2006
Q: I am a 2-digit number divisible by 19. The sum of my digit is 14. What number am I and why ? A: There aren't many two digit multiples of 19. You can write them all down and pick the only one to ...
Total Order	1/20/2006
Q: Several months ago I asked you a question that was outside of your field, but you provided me with a ... A: Every element of R2 is expressible as an ordered pair of real numbers. This can be put in 1-1 ...
Solution For Question	1/20/2006
Q: Here is mySarah's age is 2/3 of Mary's and 3/4 of Ruth's. The sum of their ages is 46 years. How old ... A: Mary is 3/2 times Sarah's age i.e. 9 sixths of Sarah's age Ruth is 4/3 times Sarah's age i.e. 8 ...
Sudoku design	1/18/2006
Q: Greeting Vijilant, How are Sudoku puzzles made? I do one on Sunday Times which is classified as very ... A: I too am a Sudoku addict, though I haven't gone into the business of designing them. I usually do ...
6-digit combination	1/13/2006
Q: I think I mean 6–number combinations, example: 12,31,45,26,10,2. Do you know of a web site where I ... A: Your print output would be at least 302558928 characters not counting line feeds and carriage ...
6-digit combination	1/13/2006
Q: How do I find every possible 6-digit combination using the numbers 0 through 50? Randy A: I'm afraid I don't understand the question. Digits are the numbers 0 to 9. If you mean how many ...
Why 10?	12/24/2005
Q: Hi, please forgive me if this question appears a little silly. I have been trying to work out why ... A: The use of base ten certainly predates the arabic system we use. The Romans used X for ten, C for a ...
equivalence relations	12/17/2005
Q: what is an eqivalence relation and an equivalence class? A: x is equivalent to y can be written x~y. For example, in the set of integers, if the relationship ...
Fermat's Last Theorem	12/16/2005
Q: What is fremat's last theorem? When was the FLT proved? By whom? A: Fermat's Last theorem states that there are no solutions in positive integers for the equation x^n ...
term paper,help needed	12/14/2005
Q: i am working on an assignment for a friend and cousin of mine on how some numbers are generated.the ... A: From the fact that the data in the two examples is identical apart from serial numbers differing by ...
what a coincidence !	12/9/2005
Q: vijilant, I thought pi was approximatly 3.14159 so why is it also used for 180 degrees ? why didn't ... A: 180 degrees is PI radians. The radian is a different measure of an angle from degrees. Suppose you ...
Please analyze my theory.	12/7/2005
Q: although this not a question that is from school, I am wonder if my theory is true. When twidding ... A: I haven't got time to answer your question fully just now -I will later (1) You will never get to ...
lub usage	12/6/2005
Q: i have been working on proving the following theorem, i have the write idea but just cant put it on ... A: I assume that you are using Q for the set of rational numbers. Consider the real line. Then the ...
primary number	11/30/2005
Q: I am Eman btayna teacher in secandary school I want to ask you How we can find the root squre of ... A: Sorry I don't know any arabic and I don't know what you mean by a primary number. If you have a ...
Thanks for the answer to my...	11/23/2005
Q: My another question is: Ismelda the witch is making a love potion. She must get the proportion of ... A: No, your reasoning is not correct because the jar does not contain pure migsweed. It is only 3/4 ...
hai the question is corrected...	11/22/2005
Q: hai the question is corrected as follows: At the edge of a pond there are five great willows in ... A: Call the numbers a,b,c,d,e. Then after the moves, the numbers are a+4,b-1,c-1,d-1,e-1. For the ...
At the edge of a pond there...	11/21/2005
Q: At the edge of a pond there are five great willows in whose branches there are in total less than 30 ... A: As stated, the problem is impossible. For instance, the fifth willow now has 4 less than before. ...
Maths	11/13/2005
Q: Question1: Joe and his dad are digging a hole for a pond in their backyard. Joe can dig a hole 7m2 ... A: Q1: Joe can dig 1/5 of the hole in an hour and his dad can dig 1/3 of it. Together they can dig ...
Math question	11/8/2005
Q: if you have the numbers 1-20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 and you have 4 ... A: You have used the word combinations, which in combinatorics means selections not counting as ...
Diophantine equation y^2+d=x^3	11/3/2005
Q: I would like to ask something about the equation y^2+d=x^3 d is a squarefree integer greater than ... A: I am afraid that this is outside my expertise, but you are on the right track with the mordell curve ...
2 induction questions	10/29/2005
Q: Let a sub k = 5 sub k-1 - 6a sub k-2 for all k >= 3, with term a1 = 3 and term a2 = 7. then a sub n ... A: This question is not really on number theory, but I will answer the first part. The equation is a ...
proofs	10/19/2005
Q: The least common multiple (lcm) is a complementary concept to gcd: we say that a positive integer m ... A: Sorry for the delay. I have been very busy lately. There are lots of ways to prove these; I will ...
math	10/8/2005
Q: i need help on magic squares A: You don't say what help you need. A magic square consists of a number of integers arranged in the ...
Maths Trivia	10/3/2005
Q: Which five figure number can you multiply by four to give the same number in reverse order. A: This is not the sort of question I answer on a regular basis. I had to put my T!-92 programmable ...
mathematical induction	9/29/2005
Q: i am not sure how to prove mathematical induction using only the propeties of natural numbers. A: I am not quite sure what you mean, but I am assuming you want to prove the inductive hypothesis. ...
number theory	9/18/2005
Q: Would you mind explaining to me what a radix fraction is? Any help would be appreciated A: The term radix refers to the number base being used, so in base ten, the radix is ten. The number ...
number theory	9/16/2005
Q: In my number theory class, which just started last week, we found the way to change base ten ... A: Assuming x,y and z are the digits in base b, the value of the number is n = xb^2+yb+z. If you want ...
primes	9/16/2005
Q: I was asked this question , but dont know where to begin. What is the smallest prime number that can ... A: You could waste a lot of time on this question, thinking it was difficult. The sum of two cubes has ...
Infinite Sum	9/15/2005
Q: and I apologize for posting in the wrong place. I have just one more question. I'm following all ... A: Even allexperts make mistakes when pushed for time, especially when out of their speciality. I did ...
Infinite Sum	9/14/2005
Q: I am attempting to solve the following summation, and I am having difficulty. infinity S ... A: This isn't a number theory question but an algebra problem to prove a result in ...
0.99999999... = 1 (is it true=? If so help me explain this)	9/8/2005
Q: A math teacher once told me that 0.99999...=1 because 1/9=0.11111..., 2/9=0.22222... and so on, so ... A: Sorry for the delayed response, but we were away for a few days and I forgot to log it as a holiday. ...
Math Theory	8/22/2005
Q: : Best wishes to the team and family that you interact with. I posted a question on August 16 of ... A: I'm not quite sure what the difficulty is now, but I repeat the calculation: S = 0.1111111... ...
Math	8/16/2005
Q: How do i prove that, while 1 over .1 is equal 10, 1 over repeated points ...1 is equal 9, in other ... A: Hola Francisco There are still eminent mathematicians who dispute whether a sum to infinity can ...
unique factorisation in Z[-1/2+/-sqrt(-19)/2]	8/8/2005
Q: it is well known that Z[-1/2+sqrt(19)i/2,-1/2-sqrt(19)i/2] (thus the algebraic integer ring in ... A: The term norm applies to any quadratic field and is useful because it helps with factorisation since ...
1/2 X 8 & 1/2 = ?/8	8/5/2005
Q: How does or why does the answer to the following calculation equal the missing term to the ... A: One of the important ideas in solving equations is that if two quantities are equal, they are still ...
Divisibility by 7 and 13	8/1/2005
Q: I have created an original method to determine if a number is divisible by 7 and 13 that is ... A: Buenas tardes Silvio I have had a good look at your site and you have answered my previous comments ...
Euclidean Algorithm	7/19/2005
Q: I have been stumped on a proof now for a week. I'm trying to prove the Euclidean Algorithm by ... A: You have assumed that gcd(a,b) = gcd(b,r). This is an easy proof, but is a necessary part of a ...
Solve For ?	7/15/2005
Q: How is the following calculation solved? 2 X 3 X ? = 4 X ? ? equals the same number.) In algebra ... A: Many great songs and poems seem to be attributed to anon, so I might be communicating with somebody ...
Table with four vars	7/9/2005
Q: Sorry, I didn't explain it well enough =) Actually the table is just a representation of what I want ... A: You must understand that this is not a number theory question. You could try using one of the free ...
Table with four vars	7/9/2005
Q: I have a table with four variables (X1, Y1, X2, Y2), and 25 different values for each. (This is ... A: You don't tell me how the table is stored. Is it a 2-dimensional array, or is it stored in cells in ...
a computer programme	7/4/2005
Q: there is a short simple programme of only a few lines long,i think it might be called the [life ... A: I just remembered I have a program on my computer called life. It is one of a suite of programs ...
Patterns In Prime Numbers	7/3/2005
Q: I was reading "Archimedes' Revenge" which states that a simple formula that generates all prime ... A: I'm afraid there is still no simple formula for generating all prime numbers, nor is there a quick ...
Proofs	6/24/2005
Q: if a^(n)-1 is a prime...1)prove that a=2 and 2)that n = prime? I proved the first part,but am ... A: I'll answer your first question and hope to get back to you with the second. You have made a good ...
Birthday paradox	6/8/2005
Q: What is the probability that at least two people from 24 people have the same birthday? (Assume 365 ... A: We calculate the probability of the birthdays all being different. The number of ways to allocate 24 ...
Learning independently	6/7/2005
Q: I just graduated from highschool. I am going into the field of Math at college. I am not sure if I ... A: You are a winner. You have triumphed over your adversity and succeeded. I am sure you will ...
Prime Numbers	5/24/2005
Q: I am trying to find out how to work out how many prime numbers there are between 1 and a million. ... A: There is definitely no quick and easy way. You have to count them or look up the answer in a book. ...
Theta(x), related to primes, zeta fn etc.	5/20/2005
Q: Let Theta(x) = SUM (of primes less than or equal to x) Log p Prove theta(x) <= (log 4)x A: The following is a standard theorem on prime number distribution. See for instance Theorem 8.5 in ...
pi, phi, square roots, and infinite decimals	5/19/2005
Q: I have recently become infatuaded with these long numbers, and have scoured the web. Now I have my ... A: What a lot of questions. There is no real point in computing constants like PI to many places of ...
Squaring	5/19/2005
Q: Sorry if this is in the wrong area or dumb or anything. I'm wondering how you could square a large, ... A: There is no generally quick method for squaring a large number, but if it has some special form, as ...
general math	5/18/2005
Q: May I know what's the difference between pure and applied math? I even heard there's a third type, ... A: Pure mathematics is the study of numbers purely for their own sake. So algebra, geometry, ...
not really a question.. about FLT	5/14/2005
Q: this is not really a question.. you might be interested in what the link below has to say on Wiles' ... A: I think the date of this article should have been April 1st. It rather reminds me of an article ...
Number Theory Proof	5/12/2005
Q: solve system of linear congruences using chinese remainder theorem. x= 2(mod7) x= 5(mod6) x= 0(mod5) ... A: The inverse of 65=30 mod 7 is 4. 430=1 mod 7 The inverse of 5*7=35 mod 6 is 5 The inverse of ...
Number Theory Proof	5/12/2005
Q: Can you help me answer and prove the following question please? What is the remainder when 5^6002 ... A: Are you familiar with Fermat's theorem? a^(p-1) is congruent to 1 mod p if p is prime and a is not a ...
arithmetic problems and number series	5/9/2005
Q: I'd appreciate if you could recommend any internet resources that's help me get ready for a test ... A: You are unfortunately in a system where they can't afford to pay exam markers properly, so they have ...
primes	5/4/2005
Q: I have read about prime numbers but nothing has ever prepared me for this question. Please help. ... A: This is very similar to Polignac's conjecture(1848),except he claimed all odd numbers could be ...
divisibility	5/4/2005
Q: I am not getting a gripe on these kind of problems I need help please. Find all pairs of positive ... A: First note that a and b must both be odd since b^2+b is even. I think this problem is like a chain. ...
Chinese Remainder Theorem	5/4/2005
Q: Hallo. Hope you are doing fine. I've been unwell thats why you haven't heard from me. I need help ... A: If you were solving the algebra equation 5x=12, you could do it by multiplying both sides by the ...
powers and mod	4/23/2005
Q: Find the last five digits of the number 9^(9^(9....(9^(9^(9^9)))...)) for 1001 9's. ie 1001 ... A: I couldn't find a way to do this analytically. I wrote a program for my TI-92 calculator and it came ...
non-axiomatic thought system	4/22/2005
Q: Please try to answer my question : is there any logical reason for mathematical problems like, ... A: We don't have to measure Pi, we can calculate it. But our calculations are based on Euclidean ...
positive integers	4/21/2005
Q: My name is Alice, a Math teacher. I challenge my mind once in a while and I got this question that ... A: No 3 successive terms are equal, so each product is either aab or abb. Each term appears 3 times in ...
Odd number out question causing headache	4/21/2005
Q: I've been given the following number sequence puzzle by a family friend who writes the things for a ... A: This is a strange one. The encyclopaedia of integer sequences at ...
Mathmetics	4/20/2005
Q: Please help my daugther with the easiest way to get this answer. Thanks N + 6 = 22 - N N = A: To solve simple equations we carry out the same mathematical operation on both sides. The operation ...
orthogonal subspaces	4/16/2005
Q: I know this isn't quite Number Theory, but was wondering if you could help me with this question? ... A: This isn't my field and I gave my books on vector spaces to an impoverished student, but I am pretty ...
GCF, LCM	4/15/2005
Q: When you say co- prime, does that mean that there is no GCF or LCM because-23 is a prime number and ... A: Yes. You've hit the nail on the head. When numbers are co-prime it means the greatest common ...
number theory	4/14/2005
Q: Here is anotherLet d and n be integers greater than 1 such that d\|n. Suppose that a is a primitive ... A: Gauss showed that the only numbers with primitive roots are 2,4,p^k and 2*p^k. The proof of this is ...
consecutive numbers	4/14/2005
Q: Let f:N\{0,1} ~N be a function defined by: f(n)=lcm[1,2,...,n] Prove that for all n,n>=2, there ... A: I don't understand what the f:N\{0,1}~N means. I think you are on the right track thinking about ...
Divisibility by seven	4/13/2005
Q: Please enter www.divisibilitybyseven.mat.br to take notice about a new divisibility by seven rule A: I am now convinced that your method does work, but doesn't involve less mathematical operations than ...
Number Theory!	4/11/2005
Q: Find all integers n, which are divisible by all the integers less than square root of n. Prove these ... A: Let's try to find the next one. 25 is no good, and numbers above 25 need a factor 5. But it ...
Brief method for determining if a number is a multiple of 7	4/11/2005
Q: I am not a Mc Auley student This article was already sent to the national library of Brazil, and ... A: I am not convinced that it is superior to the method I sent you and is not as straightforward as ...
multiplicative orders	4/11/2005
Q: I was very close to the answer I just needed someone to direct my thoughts in the right direction. ... A: In the follow up, the number a doesn't have to be a primitive root, i.e. one such that ...
multiplicative orders	4/10/2005
Q: Can you please help me with this question? Suppose d and n are integers greater than 1 such that ... A: The integers less than n and prime to it, which include a or an integer a'congruent to a mod n, form ...
complex math	4/9/2005
Q: There's a math problem I've been struggling with for several years and I was wondering if you could ... A: Sorry I can't help you a lot over this. It isn't really my field, but I did teach a course a few ...
perfect numbers	4/9/2005
Q: independent study: hope you are having a restful weekend. I wish to request you to help me ... A: To do this question, you need to understand first of all, how to calculate the sigma function ...
Fermat's Last Theorem	4/1/2005
Q: I am doing a calculus project on Sophie Germain. I have to write about a piece of mathematics that ... A: I don't think her number theory work has anything to do with calculus, but her work on elasticity ...
Number of Ways	3/31/2005
Q: I wonder if you can help with this one. (Hope my explanation makes sense!) I pick 11 different ... A: This is not a number theory question but a combinatorics one. However, since I have no teaching to ...
what does it mean that whole...	3/28/2005
Q: what does it mean that whole numbers are closed under addition but not under subtraction? A: By whole numbers I presume you mean the natural numbers 1,2,3.... Whenever you add two of these ...
limits	3/25/2005
Q: I have just purchased the book,Calculus for Dummies,and I found that by limits, it doesn't teach ... A: There are a number of ways to do this. One way is to write an infinite series for 2^x and divide it ...
divisibility by 7	3/18/2005
Q: I created a method to determine the divisibility of a number by 7, no matter the extension of the ... A: You are clearly a creative person and proud of your achievement. Your method may be a variation of ...
worded problem	3/17/2005
Q: in a group of 120 movie critics, 70 do not like Fellini, 90 like Tarkovsky, and 21 do not like both. ... A: This is a simple example is set theory. 50 like F and 90 like T. 120-21 = 99 like either or both. ...
Fibonacci question	3/17/2005
Q: I am stuck on the following problem and have no idea how to solve it. I would appreciate your help. ... A: I assume you know what is needed in a proof by induction. We need to show that the statement is true ...
Maths Puzzle	3/10/2005
Q: Can you answer this question " 999,999 to the power of 1,000,000 is a number of almost 6 million ... A: This question is about modular arithmetic. Two numbers are congruent to each other mod n if they ...
math	3/9/2005
Q: I was doing these binominal probabability distribution questions and i'm stuck on a few of them. ... A: You should be able to do most of these using the method I showed you. Q1. Work out the probability ...
math	3/9/2005
Q: a sales person has to sell a minimum of 50 cars in a month in order to get a bonus . one particular ... A: This is a statistics question and is a simple example of the Binomial distribution. The number of ...
Inverting Fractions	3/2/2005
Q: I am curious to know whether or not it is possible to invert a mixed number such as 3 1/2 without ... A: You are an investigator. You have tried to seek a method. Good. You found out that inversion ...
irrational numbers	2/28/2005
Q: Although irrational numbers are infinite nonrepeating decimals and cannot therefore be "written ... A: An expressible number means that you can describe the number by a finite string of characters. The ...
divisibility	2/26/2005
Q: why 0/0 can't be defined A: Consider the following three expressions. (a) sin(x)/x (b) sin(x)/(x^2) (c) sin(x)/(x^(1/2)) If we ...
From Pounds To Ounces	2/7/2005
Q: I want to thank you for the reply. I have a follow-up question or two. 1. Yes or No - Does the ... A: 1. Yes you have to find the number of ounces in one quarter of a pound. This is testing if you know ...
number line	1/24/2005
Q: I hope I am posting this question where it's appropriate.On the number line, 1.4 is halfway between ... A: You need to work out the mid-point for each answer. To get the mid-point you add the numbers and ...
square roots	1/14/2005
Q: What is the step by step process for calculating the square root of a given number? A: I'll do an example. However, the digits won't line up properly. The commas should be underneath one ...
unit fractions	1/10/2005
Q: what are the three unit fractions that sums up to give 1? tell me the process plz. A: I presume you mean three different unit fractions. Clearly it is easy if they do not have to be ...
Euclidean Algorithm	1/4/2005
Q: I want to thank you for your reply. However, I need some clarification. 1. I am not familiar with ... A: The process of subtracting multiples of the smallest number is just the same as taking the remainder ...
Euclidean Algorithm	1/3/2005
Q: Can the Euclidean Algorithm be used on more than two numbers to determine the greatest common ... A: There is no limit to the number of numbers. I'll choose a better example to show how it works. 16, ...
~HELP PLEASE~	12/18/2004
Q: Noah Ate Peas. Same letters stand for the same digits. Different letters stand for different digits. ... A: I don't normally answer puzzles of this sort, but I had a few minutes to spare, so here you are. ...
Well, I read in a low profile...	12/16/2004
Q: Well, I read in a low profile dictionary entry that the definable numbers can be counted (that is, ... A: This question can be answered by the classic Cantor argument. Suppose you have enumerated all ...
Proportions	12/15/2004
Q: Yes, I made an error the proportion should be 2/4 = ?/8. I know about the shortcut method, cross ... A: Let's use some algebra so that special relationships between numbers which exist in the example (all ...
Hi, What is the best you...	12/15/2004
Q: What is the best you can do when it comes to enumerating all possible mappings between sets of ... A: I am now in a position to give an authoritative answer to your question. My first response was out ...
Recursion	12/15/2004
Q: Not a McAuley student. A walkway is to be paved with a row of square tiles with 3 colors: red, ... A: This is an interesting question I haven't seen before. If we have a walkway of n-1 tiles and wish to ...
number theory	12/12/2004
Q: Is a number a multiple of itself? A: Yes. The definition of multiple of an integer is that integer multiplied by an integer including 1. ...
Fi number	12/7/2004
Q: I'm 19 year-old student of applied mathematics from Poland. My problem is that I can't cope with a ... A: You must be familiar with phi and know that the quadratic equation that defines it is f^2 = f + 1. I ...
Proportions	12/6/2004
Q: I want to thank you for the reply. I can do the mathematics, but I do not fully understand why it ... A: I appreciate that I may have used some big words which you didn't understand. You do know that when ...
Proportions	12/5/2004
Q: I know that this question my be outside of your expertise, but I hope that you can provide a reply. ... A: Remember that multiplying or dividing both elements of a proportion or both numerator and ...
please help***	12/4/2004
Q: can you please help me with the question below: thnx. Use each of these digits just once: ... A: The 5 is the first one to fix, since it cannot be a units digit. It must be the tens digit in the ...
find nth digit of 0.1234567891011...	12/4/2004
Q: Please try to explain the algorithm, in words or in a program, basic or c, as simply as possible. ... A: First we find a formula for the number of digits used to write all the numbers of up to n ...
find nth digit of 0.1234567891011...	12/1/2004
Q: I would like to know a fast way to find the nth digit of 0.1234567891011... without having to count ... A: I've been working on it. It depends what you mean by fast. I worked out that the 10^100th digit is ...
the number 'e'	11/27/2004
Q: This is my third and last question on subject. It is reported that Euler exclaimed, "God exists!", ... A: Euler's formula is e^(i*PI) + 1 =0. Incredible. All 5 of the most important mathematical numbers in ...
the number 'e'	11/26/2004
Q: appreciate your explanation of e in terms of continuously compounding interest; i.e., continuous ... A: A function that has the variable as a power or exponent is called an exponential function. The one ...
evaluating using prime factors	11/21/2004
Q: i am stuck on this question and I would very much appreciate your help, unfortunately i can't type ... A: First you need to find the prime factors. There are many methods, but the easiest one for me to ...
the number 'e'	11/9/2004
Q: Can you refer me to a layman's explanation of the number 'e'. Can such an explanation even exist? ... A: It depends what you mean by a layman. I will assume the layman has no knowledge of calculus but ...

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