Question 24
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.
(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.