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Proving divisibility

  1. Mar 2, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that n^3 - n is divisible by 6, when n is a nonnegative integer.

    3. The attempt at a solution
    Mathematical induction:

    It works for n=0
    It works for n=1 (Extra step, just in case)
    Check if it works for the (k+1)th step.

    For it to work, it must be expressible as 6x, where x is some integer.

    In other words, to prove: (k+1)^3 - k = 6x

    Can someone nudge me on this? I'm either making a mistake by calling it 6x, and maybe it should be 6k or something else...

    ...and/or, I'm just missing the algebraic skills to change LS into RS.
  2. jcsd
  3. Mar 2, 2008 #2


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    Science Advisor
    Homework Helper

    Take the lazy way!

    No no no no no!

    Far too amibitious!

    Take the lazy way!

    Just factorise [tex]n^3 - n[/tex], and you'll immediately see why 6 is always a factor! :smile:

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