VN-TST 2004 Problem 6

Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$ , let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$ . Let $a$ be greatest element from $S$ . Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $n > k$ .