RU-SPB 2022

The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$ . Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$ .

Find all pairs of nonzero rational numbers $x, y$ , such that every positive rational number can be written as $\frac{\{rx\}} {\{ry\}}$ for some positive rational $r$ .