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.
