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Thread: LCM and HCD

  1. #1
    Super Member dhiab's Avatar
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    LCM and HCD

    Find all naturels numbers a , b :

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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by dhiab View Post
    Find all naturels numbers a , b :

    Since $\displaystyle 1996 - 49\Delta^2 = 2m^2$ it is clear that $\displaystyle \Delta$ must be even. If $\displaystyle \Delta=2$ then $\displaystyle m=30$. If $\displaystyle \Delta=4$ or 6 then $\displaystyle \tfrac12(1996 - 49\Delta^2)$ is not a perfect square, and if $\displaystyle \Delta\geqslant8$ then $\displaystyle \tfrac12(1996 - 49\Delta^2)$ is negative.

    So the only possibility is that a and b have lcm 30 and gcd 2. From there, you should be able to list the possible values of a and b.
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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by Opalg View Post
    Since $\displaystyle 1996 - 49\Delta^2 = 2m^2$ it is clear that $\displaystyle \Delta$ must be even. If $\displaystyle \Delta=2$ then $\displaystyle m=30$. If $\displaystyle \Delta=4$ or 6 then $\displaystyle \tfrac12(1996 - 49\Delta^2)$ is not a perfect square, and if $\displaystyle \Delta\geqslant8$ then $\displaystyle \tfrac12(1996 - 49\Delta^2)$ is negative.

    So the only possibility is that a and b have lcm 30 and gcd 2. From there, you should be able to list the possible values of a and b.
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