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Math Help - Prove & prime

  1. #1
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    Unhappy Prove & prime

    Prove that n^13 - n is divisible by 2, 3, 5, 7 and 13 for any integer n.
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  2. #2
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    Quote Originally Posted by neosonic View Post
    Prove that n^13 - n is divisible by 2, 3, 5, 7 and 13 for any integer n.
    n^13-n=n(n^12-1).

    We will rely on Fermat's Little Theorem.
    If p|n there is nothing to prove.
    We will assume that p doth not divide n.
    That is, prove 2,3,5,7,13 divide n^12-1

    First, the case p=2 is trivial.

    If p=3 then,
    n^2=1(mod 3)
    Raise both to the power of 6,
    n^12=1(mod 3)

    If p=5 then,
    n^4=1(mod 3)
    Raise both to the power of 3,
    n^12=1(mod 3)

    If p=7 then,
    n^6=1(mod 7)
    Raise both to the power of 2,
    n^12=1(mod 7)

    If p=13 then,
    n^12=1(mod 13)

    Q.E.D.
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  3. #3
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    Thanks:)

    Thanks, it was very helpful, God bless
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