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Math Help - Prime number problem

  1. #1
    Senior Member slevvio's Avatar
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    Prime number problem

    Let a be a non zero integer and p prime.

    Show p| a^{2} => p|a.

    This is deeply perplexing me as all I can work out is gcd(p,  a^2) = p. Because p is prime, p has only 1 and p as its divisors. p divides a^2 so p is a divisor of a^2. p>1 so gcd(p,a^2) is p. If it was equal to 1 this would make my job easier but I can't work this out.

    any help would be appeciated
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  2. #2
    MHF Contributor

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    Quote Originally Posted by slevvio View Post
    Let a be a non zero integer and p prime.

    Show p| a^{2} => p|a.

    This is deeply perplexing me as all I can work out is gcd(p,  a^2) = p. Because p is prime, p has only 1 and p as its divisors. p divides a^2 so p is a divisor of a^2. p>1 so gcd(p,a^2) is p. If it was equal to 1 this would make my job easier but I can't work this out.

    any help would be appeciated
    Suppose p|a^2 and p is prime. If p\not|a then, since p is prime, \gcd(a,p)=1. By Gauss' theorem, we deduce from p|a\cdot a and \gcd(a,p)=1 that p|a, which is a contradiction.
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