Find all naturels numbers a , b :

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- Jul 16th 2009, 02:06 AM #1

- Jul 16th 2009, 05:35 AM #2
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.

- Jul 16th 2009, 06:33 AM #3

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