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Math Help - Multiplicative Inverse

  1. #1
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    Multiplicative Inverse

    This is in regard to proving that the statements (a) a and n are coprime and (c) row a of the multiplication table for Z_{n} includes all of Z_{n}

    ab = kn + 1

    "this equation implies that a and n are coprime because any common factor of a and n must also be a factor of 1

    I don't see why this is so.
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  2. #2
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    Re: Multiplicative inverse

    Let d be a common factor of a and n, so a=a_0d, n=n_0d for some integers a_0,n_0. Then

    \begin{array}{rcl} ab &=& kn+1 \\ a_0db &=& kn_0d+1 \\ \left(a_0b-kn_0\right)d &=& 1 \end{array}

    Thus d\mid1.
    Thanks from alyosha2
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  3. #3
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    Re: Multiplicative Inverse

    So because the difference is 1 we know the only factor we can pull out is 1. Good stuff. Thanks for the help.
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  4. #4
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    Re: Multiplicative Inverse

    Thank you Sylvia for solving this problem, even i had the same problem and i was also thinking of posting this but i saw Alyosha2 had already posted it and from here i got the solution of the problem and this is the best forum where we get maximum math problem solved.
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