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Thread: find remainder

  1. #1
    Junior Member
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    find remainder

    Find r when 319^566 is divided by 37.

    I can get it down to (11^2)^283*(29^2)^283.
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  2. #2
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    Hello, frankdent1!

    Are you allowed to use Modulo Arithmetic?


    Find the remainder when $\displaystyle 319^{566}$ is divided by $\displaystyle 37.$

    We find that: .$\displaystyle 319^6 \:\equiv \:-1 \pmod{37}$

    Then: .$\displaystyle 319^{566} \:=\:319^{6\cdot94 + 2} \:=\:
    (319^6)^{94}\cdot319^2$

    Hence: .$\displaystyle (319^6)^{94}\cdot319^2 \:\equiv \-1)^{94}\cdot319^2 \pmod{37}$

    . . . . . . . . . . . . . . .$\displaystyle \equiv\:319^2 \pmod{37}$

    . . . . . . . . . . . . . . .$\displaystyle \equiv\:101,761 \pmod{37}$

    . . . . . . . . . . . . . . .$\displaystyle \equiv \:11 \pmod{37}$
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