Question

If n is a is an odd positive integer show that n square minus one is divisible by 2

Answer 1

Any odd positive integer n can be written in form of 4q + 1 or 4q + 3.

If n = 4q + 1, when n2 - 1 = (4q + 1)2 - 1 = 4q2 + 2q + 1 - 1 = 2q(2q + 1) which is divisible by 2.

If n = 4q + 3, when n2 - 1 = (2q + 3)2 - 1 = 4q2 + 6q + 9 - 1 = 2(2q2 + 3q + 1) which is divisible by 2.

So, it is clear that n2 - 1 is divisible by 2, if n is an odd positive integer