Question 2
Let $a_1,a_2,\ldots,a_n$ be integers whose greatest common divisor is $1$. Let $S$ be a set of integers with the following properties:
(a) $a_i\in S$ for $i=1,\ldots,n$.
(b) $a_i-a_j\in S$ for $i,j=1,\ldots,n$, not necessarily distinct.
(c) If $x,y\in S$ and $x+y\in S$, then $x-y\in S$ too.
Prove that $S=\mathbb Z$.
Proposed by Kiran Kedlaya (credited in the supplied source).
(a) $a_i\in S$ for $i=1,\ldots,n$.
(b) $a_i-a_j\in S$ for $i,j=1,\ldots,n$, not necessarily distinct.
(c) If $x,y\in S$ and $x+y\in S$, then $x-y\in S$ too.
Prove that $S=\mathbb Z$.
Proposed by Kiran Kedlaya (credited in the supplied source).