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Math Help - inverse of modulo

  1. #1
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    inverse of modulo

    show that if a' is an inverse of a modulo m and b' is an inverse of b modulo m, then a'b' is an inverse of ab modulo m
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    a'a \equiv 1 \mod m
    b'b \equiv 1 \mod m
    hence
    1\cdot 1 \equiv a'ab'b \equiv (a'b')ab \equiv  \mod m
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  3. #3
    Super Member Random Variable's Avatar
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     aa' \equiv 1 \mod m \Rightarrow aa' = 1+mt
     bb' \equiv 1 \mod m \Rightarrow bb' = 1+ms

     ab(a'b') = aa'bb' = (1+mt)(1+ms) = 1 + mt + ms + m^{2}st = 1 + m(t+s+mst) \Rightarrow ab(a'b') \equiv 1 \mod m
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