Let $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ be a group homomorphism and suppose that $f(a,b) = (c,d)$.
Prove: $\gcd(a,b) \mid \gcd(c,d)$.
Prove: $\gcd(a,b) = \gcd(c,d)$ if $f$ is an automorphism.
I am seeking for some hint to start this problem. Although I generally have no trouble with group theoretic notions, the smallest introduction of number theory confuses me terribly. Any suggestions would be greatly appreciated.