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Find all possible values of $x, y, A, B$ if $x, y \in \mathbb Z_+$ such that $A = \frac{2y}{x(x-y)}$ and $B = \frac{(y -x)(y + 1)}{2y^2}$ are integers.

My solution: Obviously $x \neq y$ and $x, y \ge 1$. Moreover $A \neq 0$ and $B \neq 0$. Since $A$ and $B$ have to be integers their product $AB = -\frac{y+1}{xy}$ also has to be an integer. Now $GCD(y, y + 1) = 1$, so $y = 1$ since otherwise $\frac{y + 1}{y}$ wouldn't be an integer. So we can substitute $y = 1$ into $A$ and get $A = \frac{2}{x(x-1)}$. Only two values of $x$ are possible so we can check that $x = 2$ is fine both for $A$ and $B$.

Can someone please verify my solution and tell whether it is correct?

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