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Is There a Name for This Theorem?

  1. Jul 14, 2012 #1
    Is there a theorem that says when b|a2 → b|a is true for integers a and b?

    If so, what is it called?
     
  2. jcsd
  3. Jul 15, 2012 #2

    haruspex

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    I hope not, since it isn't generally true. 9|36, but not 9|6. You would need that b is a prime (or at least, has no repeated prime factors).
     
  4. Jul 15, 2012 #3
    OP, did you mean to reverse those...?

    [itex]b|a \; \rightarrow \; b|a^{2}[/itex]

    Is certainly true.
     
    Last edited: Jul 15, 2012
  5. Jul 15, 2012 #4

    AlephZero

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    But it's hardly worth calling it a theorem, since it's just a special case of ##b|a \rightarrow b|ac##.
     
  6. Jul 16, 2012 #5
    That is true whenever b is prime. You can prove it by using euclid's lemma.

    Let b be prime. Suppose b|a2. Then b|aa, and, by euclid's lemma, b|a or b|a. Hence b|a.
     
  7. Jul 16, 2012 #6
    The statement holds true whenever [itex]|\mu(b)|=1[/itex].
     
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