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Thread: Z/nZ - Exercise

  1. #1
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    Z/nZ - Exercise

    Hello,

    I don't know how to solve the last question, thanks for your help



    (a) Justify $\displaystyle \bar{11}$ ∈ (Z/17Z)*
    11 is co-prime with 17 hence 11 ∈ (Z/17Z)*


    (b) What is $\displaystyle \bar{11}^{16}$ in Z/17Z ? .
    According to Fermat's little theorem
    $\displaystyle \bar{11}^{16} \equiv 1 $ [17]


    (c) Show that $\displaystyle \bar{11}^{31}$ is a solution for the equation 11x ≡ 1 mod 17.
    $\displaystyle 11^{31} = 11^{16} \times 11^{15} = 1 \times 11^{15}$
    so $\displaystyle 11 \times 11^{15} \equiv 1 mod 17$
    $\displaystyle 11^{16} \equiv 1 [17] $
    $\displaystyle 1 \equiv 1[17] $


    (d) Deduce that $\displaystyle \bar{11}^{31}$ = $\displaystyle \bar{14}$.
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  2. #2
    MHF Contributor
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    Re: Z/nZ - Exercise

    (d)
    we need to solve the equation

    $\displaystyle 11x \equiv 1 \bmod 17$

    Show that

    $\displaystyle 11*14=154\equiv 1 \bmod 17$

    since we also have from part (c) that $\displaystyle 11*11^{31}\equiv 1 \bmod 17$

    it follows that $\displaystyle 11^{31}\equiv 14 \bmod 17$
    Thanks from topsquark
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