We have given two integers $a,b \geq 2$ and two positive integers $x \leq a$ and $y \leq b$ s.t. $(1) \;\; \frac{a}{b} - \frac{x}{y} = r$, where $r$ is the smallest positive rational number satisfying equation (1). Equation (1) is equivalent to $(2) \;\; \frac{ay - bx}{by} = r$. Hence (since $r>0$) there exists a positive integer $d$ s.t. $(3) \;\; ay - bx = d$. Equation (3) has a particular solution $(x,y)=(x_d,y_d)$ with $0 < x_d < a$. All solutions of equation (3) is given by the formulas $(x,y) = (x_d + at,y_d + bt), t \in \mathbb{Z}$. Consequently, since $0 < x \leq a$, we obtain $0 < x_d + at \leq a$, yielding $t=0$. In other words, for every $d$ equation (3) has an unique solution $(x,y)=(x_d,y_d)$. This implies that equation (3) has the unique solution $(x,y) = (dx_1,dy_1)$. Therefore $r = \frac{ay - bx}{by} = \frac{d}{b(dy_1)} = \frac{1}{by_1}$ yielding $\frac{1}{r} = by_1 \in \mathbb{N}$. q.e.d.

https://www.wiskundeolympiade.nl/phocadownload/jaarverslagen/dmo2020.pdf This is the official solution . (You can see it on page 29)