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Math Help - prime number problem

  1. #1
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    prime number problem

    please help me
    p is a prime.
    given,
    (p-1)! = p-1 (mod k)
    k = ...
    a. p+1
    b. \frac{(p-1)p}{2}
    c. \frac{(p+1)p}{2}
    d. p+1
    e. p2
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  2. #2
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    Re: prime number problem

    Hey inoid.

    If (p-1)! = (p-1) (mod k) then (p-1)! - (p-1) = 0 mod (k) which means (p-1)! - (p-1) | k or (p-1)*([p-2)! - 1] = kn for some integer n. Can you complete the rest?
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  3. #3
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    Re: prime number problem

    Consider p = 5, you will get the answer.

    Salahuddin
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  4. #4
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    Re: prime number problem

    Quote Originally Posted by chiro View Post
    Hey inoid.

    If (p-1)! = (p-1) (mod k) then (p-1)! - (p-1) = 0 mod (k) which means (p-1)! - (p-1) | k or (p-1)*([p-2)! - 1] = kn for some integer n. Can you complete the rest?
    i can't....
    please.......
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  5. #5
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    Re: prime number problem

    Ok, don't worry, I will explain.

    If p = 5, then (First of all convince yourself that 5 is a prime number. If you cant understand that, dont worry, please ask, I will explain that also).
    p - 1 = 4, (p - 1)! = 4! = 1 * 2 * 3 * 4 = 2 * 3 * 4 = 6 * 4 = 24.

    It says to find k, when (24 = 4) mod k.

    The given options are:

    (p + 1) = (5 + 1) = 6.
    (p - 1) * p / 2 = (5 - 1) * 5 / 2 = 4 * 5 / 2 = 20 / 2 = 10
    (p + 1) * p / 2 = (5 + 1) * 5 / 2 = 6 * 5 / 2 = 30 / 2 = 15.
    p^2 = 5^2 = 25.

    Take modulo for each of these numbers and check if 24 and 4 give the same moduli, with these numbers. Whichever number (among 6, 10, 15 and 25) gives same modulus, that is your answer k.

    Salahuddin
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