You are here:

# Number Theory/number theory

Question
1.)Prove that n^3-n is divisible by 6 using the exhaustion method.

2.) Prove that n^5 is divisible by 30 using the mathematical inductive method.

3.) Prove that the square of any integer is of the forms of 3k+1, bur not of the form 3k+2.

Hello Richard

(1) The exhaustion method consists of trying all the possibilies

n(mod 6)  n^3 - n (mod 6)
0         0
1         1-1 = 0
2         8-2 = 0
3         27 - 3 = 0
4         64 - 4 = 0
5         125 - 5 = 0
Every possibility gives zero, so the theorem is proved.

(2) Your second question is in error.  It should read : n^5 - n is div by 30.

Suppose the theorem is true for n = k.

Then k^5 -k is divisible by 30.

(a) It is true for k = 1  because 1^5 - 1 =0 is divisible by 30.

(b) Now we must prove (k+1)^5 - (k+1) is divisible by 5.

(k+1)^5 - (k+1) = k^5 + 5k^4 + 10k^3 + 10k^2 + 5k + 1 -k-1
We can now subtract  k^5 -k , a multiple of 30.

(k+1)^5 -(k+1) = 5k^4 + 10k^3 + 10k^2 + 5k  (mod 30)
= 5k(k^3 + 2k^2 + 2k +1) + (mod 30)
= 5k(k+1)(k^2 + k +1) (mod 30)
Now we examine the factors.  5 is a factor, 2 is a factor since we have the product of consecutive numbers.  And 3 is a factor, since if k is not a multiple of 3,
either k+1 or k^2 + k + 1 is. (exhaustion method).
So that proves it is a multiple of 30.
So by mathematical induction, the theorem is true.

This was rather harder than the average inductive proof, so don't worry because you couldn't do it.

(3) I'm going to let you try that one for yourself and get back to me if you can't do it.

Hint:  An integer is of the form 3k, 3k+1, or 3k+2.  Square each of theses and reduce (mod 3).
You will be able to prove it by exhaustion!

vijilant.

Number Theory

Volunteer

#### Vijilant

##### 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 53 years

Organizations
AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

Publications
Journal of mathematics and its applications M500 magazine

Education/Credentials
BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

Awards and Honors
State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

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.