2018 IMO Shortlist C6

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. (i) If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. (ii) If no such pair exists, we write down two times the number 0 . Prove that, no matter how we make the choices in $(i)$, operation (ii) will be performed only finitely many times. (Serbia)

令 $a$ 和 $b$ 为不同的正整数。以下无限过程发生在最初为空的板上。 (i) 如果棋盘上至少有一对相同的数字，我们选择这样的一对，并将其中一个分量增加 $a$，另一个分量增加 $b$。 (ii) 如果不存在这样的对，我们写下两倍的数字 0 。证明，无论我们如何在 $(i)$ 中做出选择，操作 (ii) 都只会执行有限次。 (塞尔维亚)