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Expert: Robi Bhattacharjee - 9/1/2008


Question
Hi,
I need to prove by induction that 5^n+3 is divisible by 4 for all nЄZ+

Answer
Hello Taxir,

Great question! Well in mathematical induction, you first verify it works for n = 1. So 5^1 + 3 = 8 which is divisible by 4.

Now let us assume that 5^n + 3 is divisible by 4 for n = j. If we prove that with our assumption being true that it works for j + 1, then we are done.

So 5^j + 3 is divisible by 4

now 5^(j+1) + 3 = 5 * 5^j + 3 = (1+4)5^j + 3 = 5^j + 4*5^j + 3

= 5^j + 3 + 4*5^j

now 5^j + 3 is divisible by four by our assumption, and 4*5^j is divisible by 4 since 4 is divisible by 4. This means that the whole expression is divisible by 4 and that means 5^(j+1) + 3 is divisible by 4.

So we have proved that if 5^j + 3 is divisible by four then 5^j+1 is divisible by four. This means let j = 1. Then it works for two since it works for one. Let j = 2, it works for 3. So by induction, it works for all n.

I hope this helped,

Feel free to ask if anything was unclear,

Robi  

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Robi Bhattacharjee

Expertise

I can answer a variety of questions on mathematics. Questions on trigonometry, calculus(preferably single variable), algebra, geometry, and number theory will be answered. I cannot answer questions on abstract branches of mathematics such as group theory. I also cannot answer questions on statistics. In number theory, I can answer questions on congruences, prime numbers, units, functions, and the riemann-zeta function.

Experience

I have studied advanced math my entire life. I started calculus in sixth grade. I have attended numerous math competitions and I am attending math organizations such as the San-Diego math circle. Also, this year I have been invited to the USAMO which is a prestigious math competition (Every year the USAMO invites 500 students from across the USA to participate in this competition. The top 6 go to represent the USA in the International Math Olympiad).

Organizations
I am in the San Diego Math Circle

Education/Credentials
I am entering high school and have received a perfect score and the STAR test 5 times in a row. I also have gotten recognitions in the AMC 10, AIME, Math Counts, and ARML. Additionally, I have won the San Diego Math Olimpiad twice in a row.

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