Question 6
Let $S$ be a nonempty set of positive integers whose smallest and largest elements are coprime. For each natural number $n$, let $S_n$ be the set of all natural numbers expressible as a sum of at most $n$ elements of $S$, with repetitions allowed. The sum of zero elements is 0. If $a=\max S$, prove that there are a positive integer $k$ and an integer $b$ such that
$$|S_n|=an+b\qquad\text{for every }n>k.$$
Here $|X|$ denotes the number of elements of $X$, and natural numbers in the definition of $S_n$ include 0.
$$|S_n|=an+b\qquad\text{for every }n>k.$$
Here $|X|$ denotes the number of elements of $X$, and natural numbers in the definition of $S_n$ include 0.