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Math Help - Need to finish this question by matlab? please help

  1. #1
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    Need to finish this question by matlab? please help

    Let n=391=17*23. Show that 2n-11 (mod n). Find an exponent j>0 such that
    2j1 (mod n)
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    If those are exponents you might want to be a little more clear.

    Are you saying :

    Let n=391=17\times23. Show that 2^{n-1} \not\equiv 1 \mod n. Find an exponent j>0 such that
    2^j\equiv 1 \mod n .
    ?

    If so, then by no means do you need a computer to solve this problem.
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  3. #3
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    Yes, this question need to solve by Matlab, but I am not good for matlab. Do you have any ideas??


    thanks for help!!
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Use Euler's theorem, which states that for any integer a, relatively prime to n,

    a^{\phi(n)} \equiv 1 \mod n

    where \phi is Euler's phi function.

    So, by Euler's theorem, with \phi(391)=(17-1)(23-1)=352, you have 2^{352} \equiv 1 \mod 391.

    For the first part you can use the Chinese Remainder Theorem. You have :

    2^{391} \equiv (2^{17})^{23} \equiv 2^{23} \equiv 2^{17}2^6 \equiv 2^7 \equiv 128 \equiv 9 \mod 17

    so it's impossible that 2^{391} \equiv 2 \mod 391.

    Matlab is for engineers.
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