Proof with a subset of linear combinations of integer numbers
Assume that the set S is a subset of integer numbers. Assume it has a property that if numbers x and y belong to S then their integer linear combination nx + my also belongs to S. Prove that either S consists from one number zero or there is a positive number d such that any other number in S is proportional to d.
I already proved that it consists only of zero when x=y=0 and that otherwise there is a positive number that belongs to S, but I can't figure out how to prove the last part. I'm pretty sure that d must be the gcd, but I'm not sure if I have to prove that too.