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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.