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Math Help - Number of pairs with give GCD and LCM (proof)

  1. #1
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    Question Number of pairs with give GCD and LCM (proof)

    Hello, I would like a proof for the following algorithm, please.

    In order to find the number of pairs with a given GCD (greatest common divisor) and LCM (least common multiple), I find the number n of prime factors in LCM/GCD. The number of pairs is equal to 2^n.


    Example: GCD=2 and LCM=120

    LCM/GCD = 60 = 2^2 * 5 * 3 therefore we have 2^3 (we have 3 prime factors) pairs.
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  2. #2
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    Re: Number of pairs with give GCD and LCM (proof)

    Hi,
    Your statement is not quite precise. The attachments give a precise statement and proof.

    Number of pairs with give GCD and LCM (proof)-mhfnumbertheory4aa.png

    Number of pairs with give GCD and LCM (proof)-mhfnumbertheory4b.png
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  3. #3
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    Re: Number of pairs with give GCD and LCM (proof)

    Thanks a lot for your help, I understand it now
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