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