Question

if n is odd integer then show that n2-1 is divisible bye 8​

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 = 16q2 + 8q + 1 - 1 = 8q(2q + 1) which is divisible by 8.

If n = 4q + 3, when n2 - 1 = (4q + 3)2 - 1 = 16q2 + 24q + 9 - 1 = 8(2q2 + 3q + 1) which is divisible by 8.

So, it is clear that n2 - 1 is divisible by 8, if n is an odd positive integer.