Number of integer solutions to the diophantine equation

You already made good progress. When you got $$x=\frac{6y}{y-6}$$ you should write it as $$x=\frac{6y}{y-6}=\frac{6(y-6)+36}{y-6}=6+\frac{36}{y-6},$$ and since $x$ is an integer we must have that $36/(y-6)$ is an integer as well, in other words $(y-6)\mid 36$. This leaves finitely many values to check, which I'm sure you can do.

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Linear Algebra

Algebra Precalculus

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