1) Find all prime numbers that divide 50!
2) If p and are both prime numbers, prove that is also prime.
3) (a) If p is a prime, and , prove that in the AP a, , , , , 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 denote the nth prime. For , show that .
Hint: Use induction and Bertrand's conjecture.
5) Prove that for every , there exists a prime p with .