The question:
Let $a,b \in \mathbb{Z}$ with $a$ and $b$ not both zero, and let $d = (a,b)$. Prove that $d$ is the least positive integer linear combination of $a$ and $b$; ie ,for all $c \in \mathbb{Z}^{+}$, if $c$ is an integer linear combination of a $a$ and $b$, then $d \leq c$
I tried finding similar examples of this proof on stack exchange and google but found nothing.
I suppose you already know that $d$ itself is an integer combination of $a$ and $b$. So you only need to prove it is the smallest. Well, suppose $c>0$ satisfies $c=ma+lb$ when $m,l\in\mathbb{Z}$. We know that $d|a$ and $d|b$. You can easily check that it implies $d$ must divide any integer combination of $a$ and $b$. Hence $d|c$ and since both are positive $d\leq c$.
