CN-TST 2019 Problem 12

Given coprime positive integers $p,q>1$ , call all positive integers that cannot be written as $px+qy$ (where $x,y$ are non-negative integers) bad, and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$ . Prove that there exists a positive integer $n$ , such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$ .