Question

Prove that there is a prime between n and n factorial

Answer 1

Let k = n! - 1. Well, since all numbers from 2 to n divide n!, none of these divides k. This means that k either a prime , in which case we are done, or there exists a prime p > n which divides n! - 1. In the latter case, p is between n and n factorial and p is prime, so the proof is complete.