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Thread: Some application of Euler's Theorem probably

  1. #1
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    Some application of Euler's Theorem probably

    $\text{Find the smallest }n \in \mathbb{N^+} \ni \text{ the last two digits of }3^n \text{ written in base }143 \text{ are }\{0,~1\}$

    I know you want to show that

    $3^n \equiv 1 \pmod{143^2}$

    and that

    $3^{\varphi(143^2)} \equiv 1 \pmod{143^2}$

    but $\varphi(143^2)=17160$ is not the smallest value that satisfies this, 195 is.

    How to derive 195?
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  2. #2
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    Re: Some application of Euler's Theorem probably

    if we can show that $a=5$ is the smallest positive integer such that

    $3^a\equiv 1 \pmod {11^2} $

    and $b=39$ is the smallest positive integer such that

    $3^b\equiv 1 \pmod {13^2} $

    then it will follow that $n=5*39=195$ is the smallest positive integer such that

    $3^n\equiv 1 \pmod {143^2} $
    Thanks from romsek and topsquark
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