USAMO 2004 Problem 2

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties:

(a) For $i=1, \dots, n$ , $a_i \in S$ . (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$ . (c) For any integers $x,y \in S$ , if $x+y \in S$ , then $x-y \in S$ .

Prove that $S$ must be equal to the set of all integers.