Question

Prove that for all integer m and n, if m-n is even then m^3-n^3 is even.

Answer 1

Let $m,n$ be any two integers, and assume $m-n$ is even. (This would mean either both $m,n$ are even or odd, but that's not important.)

We have

$m^3-n^3=(m-n)(m^2+mn+n^2)$

and the parity of $m-n$ tells us $m^3-n^3$ must also be even. QED