Question 16
Let $Z^+$ denote the set of all positive integers and $P$ denote the set of all prime numbers. For subsets $A$ and $S$ of $Z^+$, $A$ is called $S$-proper if there exists a positive integer $N$ such that for all $a \in A$ and integer $b$ with $0 \le b \lt a$ there exist not necessarily distinct elements $s_1, s_2, \dots, s_n$ of $S$ satisfying the conditions $b \equiv s_1 + s_2 + \dots + s_n \pmod a$ and $1 \le n \le N$. Find a subset $S$ of $Z^+$ for which $P$ is $S$-proper but $Z^+$ is not.