Question

Show that for any integer x, the greatest common divisor of x+1 and x-1 is either 1 or 2.

Answer 1

Assume otherwise - there exists a p > 2 dividing x-1 and x+1.

Then x-1 = kp and x+1 = lp, where k, l are integers, k < l.

lp - kp = 2, so (l-k)p = 2, which is a contradiction, because p > 2. QED