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Math Help - Fermat's little theorem

  1. #1
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    Fermat's little theorem

    Using Fermat's little theorem, how do i find the solutions of 7x=12(mod 17)??

    i have NO clue!!!
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  2. #2
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    Since 17 is a prime, and 17 does not divide 7, by Fermat's little theorem, we see that

    7^{17-1} = 7^{16} \equiv 1 \bmod{17}

    Then

    7^{16} = 7 \cdot 7^{15} \equiv 1 \bmod{17} \Rightarrow 7 \cdot 7^{15}\cdot 12 \equiv 12 \bmod{17}.

    So

    x = 7^{15} \cdot 12 \text{, or } x=7^{15} \cdot 12 \equiv 9 \bmod{17}.

    Hope that helps!
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