### Divisibility & Induction

Suppose $x$, $y$, $z$ are positive integers satisfying the equation $\frac{1}{x} - \frac{1}{y} = \frac{1}{z},$ and let $h$ be the highest common factor of $x$, $y$, $z$.

Prove that $hxyz$ is a perfect square.

Prove also that $h(y-x)$ is a perfect square.

Source: British Mathematical Olympiad, Paper 2, 1998, Question 3.