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Math Help - modular equations?

  1. #1
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    modular equations?

    I am just wondering how do you solve this without a calculator 5^55mod(12)=?

    Thx in advance
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: modular equations?

    Quote Originally Posted by ponk View Post
    I am just wondering how do you solve this without a calculator 5^55mod(12)=?

    Thx in advance
    Welcome to MHF, ponk!


    Here's a standard way to reduce the expression.

    \begin{align*} 5^{55} \textrm{ mod }12 &=5^{1+2 \times 27} \textrm{ mod }12 \\ &=5 \times (5^2)^{27} \textrm{ mod }12 \\ &= 5 \times (25  \textrm{ mod }12)^{27} \textrm{ mod }12 \\ &= 5 \times (1)^{27} \textrm{ mod }12 \\ &= 5 \end{align*}

    In this case we were done rather quickly since 25 \textrm{ mod }12 = 1, but otherwise the process could have been repeated.

    For more complicated expressions Euler's theorem comes in handy.
    Last edited by ILikeSerena; January 1st 2012 at 06:09 PM.
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  3. #3
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    Re: modular equations?

    thx a bunch
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