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Math Help - Modulo Question: Proof

  1. #1
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    Modulo Question: Proof

    Prove that there is no integer x satisfying ax 1(mod m) if a is a divisor of m and a is neither 1 nor m.

    All i did:
    ax= 1 + rm where r is integer
    Then:
    x=1/a +rm/a
    dont know how to continue.

    Any help will be appreciated.
    Thanks in advance

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  2. #2
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    Quote Originally Posted by firebio View Post
    Prove that there is no integer x satisfying ax 1(mod m) if a is a divisor of m and a is neither 1 nor m.

    All i did:
    ax= 1 + rm where r is integer
    Then:
    x=1/a +rm/a
    dont know how to continue.

    Any help will be appreciated.
    Thanks in advance

    a\mid m\Longrightarrow m=ab ; if we had xa=1\!\!\!\pmod m , then we'd get xa=1+km=1+akb\Longrightarrow a(x-kb)=1\Longrightarrow a=\pm 1 , which is not the case by assumption.

    Tonio
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