Number Theory Problems

1. Show that if a is an integer greater than 1, then a does not divide 2ª-1 (Putnam 1972).

2. Prove that (n+m)ª = nª + mª mod(a), where a is any prime and n and m are any integers. Conversely, prove that if a is a divisor of all coefficients of the expansion of (n+m)ª, except the first and the last, then a is a prime or a=1.

3. Prove that for any set of n integers, there is a subset of them whose sum is divisible by n.

4. For any positive integer n, prove that there exist n consecutive integers each of which contains a repeated prime factor.

5. If S is any set of n+1 integers selected from 1,2,3,…,2n+1, prove that S contains two relatively prime integers.

6. Prove that among any subset of n+1 integers selected from the set {1,2,3…,2n} there are two elements a and b such that a is a multiple of b. (Erdos problem).