Question 22
Let $a$, $b \ge 2$ be positive integers with $\gcd(a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b} - \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.