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Math Help - Greatest common divisor

  1. #1
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    Greatest common divisor

    Hi.
    Can anybody prove that?

    Let n>4

    GCD(n, (n-1)!)=1 \Longleftrightarrow{} n its prime.

    and

    if n its not prime \Longrightarrow{} GCD(n, (n-1)!)=n

    Thanks
    Last edited by asub1; December 3rd 2008 at 02:02 PM.
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  2. #2
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    Quote Originally Posted by asub1 View Post
    GCD(n, (n-1)!)=1 \Longleftrightarrow{} n its prime.
    This should be clear. If n is prime then 1,2,3...,n-1 show no non-trivial factors with n. Thus, (n,(n-1)!)=1.

    if n its not prime \Longrightarrow{} GCD(n, (n-1)!)=n
    If n is not prime then n=ab where 1<a,b<n. Therefore, we are able to find a,b among 1,2,...,n-1.
    Thus, n|(n-1)! and so (n,(n-1)!)=n.
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  3. #3
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    Hi!


    If n is not prime then n=ab where 1< a,b <n. Therefore, we are able to find a, b among 1, 2, ..., n-1.
    Thus, n|(n-1)! and so (n, (n-1)!)=n.
    What about a=b?
    n=aa where 1<a, a<n. we are able to find a among 1, 2, ..., n-1, but... and the other a?

    Can you prove?

    Thanks
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