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my idea is to factor the LCM to create a pool of options

Maybe algebra with a quadratic equation?

maybe. but number theory methods might be perfered

\[\large x \left( 1640-x \right)=8400n\]
where n is a natural number

hmm, the "n" seems interesting

\[x=820 \pm \sqrt {820^2-8400n}\]

i assume the under radical needs to remain 0 or greater?

needs to be a perfect square

lol, yeah i spose that would have to be a major caveat seeing the x needs to be an integer :)

\[\large x=820 \pm 400\sqrt{1681-21n}\]

i think one more condition was such that x and y were both positive values