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Math Help - 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 319^{566} is divided by 37.

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

    Then: . 319^{566} \:=\:319^{6\cdot94 + 2} \:=\:<br />
(319^6)^{94}\cdot319^2

    Hence: . -1)^{94}\cdot319^2 \pmod{37}" alt="(319^6)^{94}\cdot319^2 \:\equiv \-1)^{94}\cdot319^2 \pmod{37}" />

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

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

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