Let P be a prime and let n be a positive integer Prove that

Solution

Let n be an integer. Suppose a | n and a | (n + 1). Then as = n and at = n + 1 for some s, t Z. But 1 = (n + 1) n = at as = a(t s) therefore a divides 1. Therefore the only common divisors of n and n + 1 are 1 and 1. In particular, a prime cannot divide both n and n + 1.

REPLACE N WITH P