1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    User Avatar
    Science Advisor
    Homework Helper

    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].
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook