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,
Regards,
Nalin Pithwa

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