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Math Help - Prime Theorem-Proof?

  1. #1
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    Prime Theorem-Proof?

    P\a
    Last edited by thegarden; December 2nd 2008 at 08:26 AM.
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  2. #2
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    Quote Originally Posted by thegarden View Post
    Proof required for this theorem:
    Let p be prime, p|ab implies p|a or p|b.
    Thanks.
    Surely p isn't a prime then?
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  3. #3
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    I think we can assume p is prime
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  4. #4
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    If p is prime, then p does not divide a. And p does not divide b. By definition of being prime. (Unless a and b are 1 and p).
    So proving that p divides a or b would be rather difficult, no?

    Are you sure you've got the question right?

    a and b are integers, I'm assuming...
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  5. #5
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    Quote Originally Posted by thegarden View Post
    Proof required for this theorem:
    Let p be prime, p|ab implies p|a or p|b.
    Thanks.
    This theorem is correct. (" p|a" means that p divides a, or that a is a multiple of p, not the contrary)

    It results from Gauss Theorem: We assume that p|ab. If p|a, we are done. Suppose p\not|a. Then a and p are relatively prime.
    (If d divides a and p, then d=1 or d=p because p is prime, hence necessarily d=1 since p\not|a; hence 1 is the only positive common divisor of a and p)
    Because p|ab, you deduce from Gauss Theorem that p|b. qed
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