MathDB

Let

a

and

b

be distinct positive integers. The following infinite process takes place on an initially empty board.
[*] 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

. [*] If no such pair exists, we write 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.
Proposed by [I]Serbia[/I].

Problem Statement