Homework problems for 5 Nov 2011:
- Let a1, a2, … be a series recursively defined as follows: a1 = 20, a2 = 11, and for n ≥ 1, an+2 is the remainder when an+1 + an is divided by 100. What is the remainder when is divided by 8?
- (IMO 1962 Q1) Find the smallest positive integer with the following properties:
- in decimal representation, its last digit is 6;
- if we move this digit to the front of the number, we get 4 times the original number.
Homework problems for 12 Nov 2011:
- Prove that if a, n are integers and n > 0, then whenever , we have .
- Classify the solutions of , where x, y and z are positive integers.
- Prove that if n =3k, then n | (2n + 1). In particular, there are infinitely many n such that n | (2n + 1). Can you find an n with at least 2 distinct prime factors such that n | (2n + 1)?
- (IMO 1971 shortlisted) Find all integer solutions to .
- Suppose n is a positive integer such that n | (2n + 1). Prove that n = 1 or n is a multiple of 3.
[ Coming attractions : quadratic residues… stay tuned! 🙂 ]