mmm

x^2 - 1


Plug any odd integer other than one into x, and the result is always divisible by 8. Why?

It works for 1 as well.

Substitute 2n+1 (odd for any integer n) for x, you get 4n(n+1). Either n or n+1 is divisible by 2 (because n is an integer), so the whole expression is always divisible by 8. QED.