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If a^2 divides b^2, a divides b

  1. Sep 11, 2016 #1
    1. The problem statement, all variables and given/known data
    If a^2 divides b^2, then a divides b
    Also
    If a^2 divides b^3, then a divides b

    2. Relevant equations


    3. The attempt at a solution
    For the first question, if a^2 divides b^2, then b^2=(a^2)c where c is some integer
    c=(b^2)/(a^2)
    c=(b/a)^2
    The answers in the back say that it is enough to show that c is a perfect square, but I don't see how that's sufficient. If c is a perfect square, then c is an integer, which implies that a^2 divides b^2, which we already knew.

    No idea where to go for the second question. The same strategy doesn't seem to work.
     
  2. jcsd
  3. Sep 11, 2016 #2
    It might help to try to establish this fact: the square root of an integer is either an integer or an irrational number.
     
  4. Sep 11, 2016 #3

    PeroK

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    A more general approach to those problems would start with:

    Let ##p## be a prime divisor of ##a## of order ##n##.
     
  5. Sep 11, 2016 #4

    PeroK

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    You could try to find a counterexample!
     
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