<s>[INST] Determine all pairs of $(x,y)$ such that $\frac{x^2y+x+y}{xy^2+y+11}$ is an integer Here is a problem that I got:

Determine all pairs of positive integer $(x,y)$ such that $\dfrac{x^2y+x+y}{xy^2+y+11}$ is an integer.

My attempt:
First notice that $(11, 11)$ is a solution. And if $\dfrac{x^2y+x+y}{xy^2+y+11}$ is an integer, then $xy^2+y+11|x^2y+x+y$. Hence $x^2y-xy^2+x-11|xy^2+y+11$.
And then I'm stuck from here. Can anyone help me? Thanks.
 [/INST] Hint: Notice that $$xy^2+y+11 \mid x(xy^2+y+11)-y(x^2y+x+y)=11x-y^2$$
Then you have three cases to check: $11x-y^2=0$, $11x-y^2<0$, and $11x-y^2>0$.
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