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Math Help - Modulo proof

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    Modulo proof

    Let n E Z. and supposed that 5 does not divide n. Prove that n^4 is congruent to 1 mod 5.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by kel1487 View Post
    Let n E Z. and supposed that 5 does not divide n. Prove that n^4 is congruent to 1 mod 5.
    Nothing fancy here. For example,
    n \equiv 1~\text{ (mod 5)}
    so
    n^4 \equiv (1)^4 \equiv 1~\text{ (mod 5)}

    Similarly for n \equiv 2~\text{ (mod 5)}
    n^4 \equiv (2)^4 \equiv 16 \equiv 1~\text{ (mod 5)}

    etc.

    -Dan
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    Thanks!! I was thinking too much into it!!
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  4. #4
    Newbie Catherine Morland's Avatar
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    If n is not divisible by 5, then one of n+1, n-1, n+2, n-2 must be divisible by 5.

    Hence the product (n+1)(n-1)(n+2)(n-2)=(n^2-1)(n^2-4) must be divisible by 5.

    Note however that n^2-4\equiv n^2+1\pmod{5}.

    Hence (n^2-1)(n^2+1)=n^4-1 is divisible by 5; in other words n^4\equiv1\pmod{5}.

    In general, Fermat's little theorem states that if p is prime and p does not divide n, then n^{p-1}\equiv1\pmod{p}.
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