Middle European Mathematical Olympiad 2022 Problem I-4

Initially, two positive integers $a$ and $b$ with $a \neq b$ are written on a blackboard. At each step, Andrea picks two numbers $x$ and $y$ on the blackboard with $x \neq y$ and writes the number

$$\gcd(x, y) + \operatorname{lcm}(x, y)$$

on the blackboard as well. Let $n$ be a positive integer. Prove that, regardless of the values of $a$ and $b$, Andrea can perform a finite number of steps such that a multiple of $n$ appears on the blackboard.

Remark. If $x$ and $y$ are two positive integers, then $\gcd(x, y)$ denotes their greatest common divisor and $\operatorname{lcm}(x, y)$ their least common multiple.