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Math Help - divisibility proof

  1. #1
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    divisibility proof

    The question I am working on is
    Prove that a/b if and only if a^3/b^3.

    I know that if p/a^n then p/a...but how can i apply that here..can i make a prime on my own despite the question did not say?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by empressA View Post
    The question I am working on is
    Prove that a/b if and only if a^3/b^3.

    I know that if p/a^n then p/a...but how can i apply that here..can i make a prime on my own despite the question did not say?
    I'm pretty sure we can go through with bi-implications this way.

    a|b \Leftrightarrow b = ka for some k \in \mathbb Z

    \Leftrightarrow b^3 = k^3a^3

    \Leftrightarrow a^3|b^3 since k \in \mathbb Z \implies k^3 \in \mathbb Z

    and done.


    Now if that makes you uncomfortable, and you want to do both directions, change all the above \Leftrightarrow's to \implies's and consider that your proof of the forward direction.

    For the backwards direction, you will need to work a little harder. Personally, I'd do a proof by contradiction, employing the division algorithm.
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