For a positive integer n, define n factorial to be the integer n! = n(n − 1)(n − 2)· · · 1.
(a) Suppose 1 ≤ k ≤ n. What are the quotient and remainder when N = n! + 1 is divided by k? Explain.
(b) Explain why part (a) implies that N has a prime divisor greater than n.
(c) Explain why part (b) implies that there are infinitely many prime numbers. (Note: if there are only finitely many prime numbers, then there is a largest prime.)