# Thread: LCM and HCD

1. ## LCM and HCD

Find all naturels numbers a , b :

2. Originally Posted by dhiab
Find all naturels numbers a , b :

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

3. Originally Posted by Opalg
Since $1996 - 49\Delta^2 = 2m^2$ it is clear that $\Delta$ must be even. If $\Delta=2$ then $m=30$. If $\Delta=4$ or 6 then $\tfrac12(1996 - 49\Delta^2)$ is not a perfect square, and if $\Delta\geqslant8$ then $\tfrac12(1996 - 49\Delta^2)$ 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.
Hello : Thank you. are you the details ?

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# hcd and lcm

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