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Math Help - Proof...having trouble in one direction.

  1. #1
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    Proof...having trouble in one direction.

    1) Let p be prime, m be a positive integer
    Show that if a is an integer, then gcd(a,p^m)=/=1 if and only if p divides a.

    I think I got the first way:
    So first I suppose the gcd (a,p^m)=/=1. So a and p^m have a common factor greater than one. Call it k. So k divides p^m and k divides a. Since k divides p^m, k divides p. So p=lk for some integer l and a=mk for some integer m. So k=p/l means a=m(p/l)=p(m/l), and since m/l is an integer, p divides a.

    For the second way, I started, but got stuck, and maybe it is all wrong:

    Now suppose p divides a. Then a=mp for some integer m. Suppose that gcd(a,p^m)=1. Then a and p^m have no common factors besides 1. Stuck. Thanks
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  2. #2
    Member Black's Avatar
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    Quote Originally Posted by twittytwitter View Post
    Since k divides p^m, k divides p.
    But how can this be true if p is prime?

    If \text{gcd}(a,p^m)=k>1, then k|p^m \Longrightarrow k=p^n for n \le m. Therefore, p|k \Longrightarrow p|a.

    If p|a, then \text{gcd}(a,p)=p. So \text{gcd}(a,p^m) \ge p.
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