I know that $\gcd(a,b)$ divides $a$ and $b$, and must also then divide $(a)(t)$ ($t$ being some integer). This makes sense to me, but how do I prove it? It seems that the addition of $(a)(t)$ is a continuation of the linear combination of $\gcd(a,b) = av + bu$ for some $v$, $u$ being elements of $\mathbb{Z}$.
Any help?