What are the number of integer solutions of $xy - 6 (x+y)=0$ with $x\leq y$ is ?
Equation $xy - 6 (x+y)=0$ can also be written as $1/x + 1/y = 1/6$
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The equation can also be written as $(x-6)(y-6)=36$. So $x-6$ and $y-6$ are integers, not necessarily positive, whose product is $36$.
How many divisors, not necessarily positive, does $36$ have? Then you will have to take care of the $x\le y$ constraint. This can be more or less done by symmetry.