A random collection of number theory problems for RMO and CMI training

1) Find all prime numbers that divide 50!

2) If p and p^{2}+8 are both prime numbers, prove that p^{3}+4 is also prime.

3) (a) If p is a prime, and p \not|b, prove that in the AP a, a+b, a+2b, a+3b, \ldots, every pth term is divisible by p.

3) (b) From part a, conclude that if b is an odd integer, then every other term in the indicated progression is even.

4) Let p_{n} denote the nth prime. For n>5, show that p_{n}<p_{1}+p_{2}+ \ldots + p_{n-1}.

Hint: Use induction and Bertrand's conjecture.

5) Prove that for every n \geq 2, there exists a prime p with p \leq n < 2p.

More later,
Nalin Pithwa

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