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Math Help - Prime Numbers and Divisibility

  1. #1
    Super Member Aryth's Avatar
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    Prime Numbers and Divisibility

    What is the remainder when 5!25! is divided by 31?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Aryth View Post
    What is the remainder when 5!25! is divided by 31?
    5! 25! \equiv 5! 23! (-7) (-6) \equiv 5! 23! 11 \mod 31

    ................. \equiv 5! 22! 5 \mod 31

    ................. \equiv 5! 21! 17 \mod 31

    etc

    CB
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  3. #3
    Senior Member JaneBennet's Avatar
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    Or use Wilson’s theorem.

    30!\equiv-1\pmod{31}

    30\times29\times28\times27\times26\equiv(-1)(-2)(-3)(-4)(-5)\pmod{31}\equiv-5!\pmod{31}

    \therefore\ 5!25!\equiv-30!\pmod{31}\equiv1\pmod{31}
    Last edited by JaneBennet; April 10th 2009 at 01:59 PM.
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