# Direct proof using definiton of even

1. Feb 19, 2009

### cmajor47

1. The problem statement, all variables and given/known data
Prove that for all integers n and m, if n-m is even then n3-m3 is even.

2. Relevant equations
Definition of even: n=2k

3. 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?

2. Feb 19, 2009

### HallsofIvy

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

Last edited by a moderator: Feb 19, 2009
3. Feb 19, 2009

### cmajor47

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