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Direct proof using definiton of even

  1. Feb 19, 2009 #1
    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 [tex]\in[/tex] 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. jcsd
  3. Feb 19, 2009 #2

    HallsofIvy

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    How about using the fact that x3- y3= (x- y)(x2+ xy+ y2)?
     
    Last edited: Feb 19, 2009
  4. Feb 19, 2009 #3
    Thanks so much, I realized how to do the proof with that help.
     
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