Direct proof using definiton of even

[cmajor47]

Homework Statement

Prove that for all integers n and m, if n-m is even then n³-m³ is even.

Homework Equations

Definition of even: n=2k

The Attempt at a Solution

Proof: Let n, m $$\in$$ Z such that n-m=2k
n-m=2k
n=2k+m
m=-2k+n
n³-m³=(2k+m)³-(-2k+n)³

I did all of this algebra out but I didn't think that it worked in showing that n³-m³ is even. Am I doing the proof wrong?

 

[HallsofIvy]

How about using the fact that x³- y³= (x- y)(x²+ xy+ y²)?

 

[cmajor47]

Thanks so much, I realized how to do the proof with that help.