(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares. (b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.