Question 177
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.
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.