CN-TST 2018 Problem 9

Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$ . Determine the smallest $n$ so that such operation can go on infinitely.