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Wilson's Theorem remainder

  1. Apr 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the remainder when 34! is divided by 37.

    2. Relevant equations

    Wilson's Theorem

    3. The attempt at a solution

    I understand that (p-1)! = (-1)(mod p) and that (p-2)! = (1)(mod p). I don't understand how to apply this to (p-3)! though.
     
  2. jcsd
  3. Apr 19, 2009 #2
    you know that (p-2)! = 1 (mod p). So (p-3)!*(p-2) = 1 (mod p). In this situation, 34!*35 = 1 (mod 37). Call 34! 'x' and then solve 35x = 1 mod 37, which has a unique solution since gcd(35,37) = 1.
     
  4. Apr 19, 2009 #3
    So do you do..

    1 = 35x
    1 = (-2)x
    1 = (-2)(-18)

    R = -18 + 37 = 19 ??
     
  5. Apr 20, 2009 #4

    Dick

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    35*19 isn't 1 mod 37. Don't you mean 1=(-2)(-19)? It's easy enough to check your answers with a quick calculation.
     
  6. Apr 20, 2009 #5
    You're right. Thanks for the help.
     
  7. Apr 24, 2009 #6
    I'm trying to find 33! / 37 now.

    I have gotten to (-3)x = 18 (mod 37)... but I can't figure out what x is.
     
  8. Apr 24, 2009 #7

    Dick

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