Question 126
Let $n \gt 1$ be a given integer and $A$ be an infinite set of positive integers satisfying: for any prime $p \nmid n$, there exist infinitely many elements of $A$ not divisible by $p$. Prove that for any integer $m \gt 1$, $(m, n) = 1$, there exists a finite subset of $A$ whose sum of elements, say $S$, satisfies $S \equiv 1 \pmod m$ and $S \equiv 0 \pmod n$.