Question 140
In the beginning, there are two positive integers on a blackboard. On each step, one chooses numbers $a$ and $b$ such that $a \le b$ from the numbers on the blackboard in all possible ways (equality means that one may take the same number twice), finds all corresponding sums $a+b+\gcd(a,b)$ and replaces all the numbers on the blackboard instantly with these sums. Prove that at some step at least one number will occur more than once on the blackboard.