Here is my proof of $\gcd(a,b)=ax+by$ for $a, b, x, y \in \mathbb{Z}$. Am I doing something wrong? Are there easier proofs?
$a,b \in \mathbb{Z}, g=\gcd(a,b)$ and suppose $g \neq ax + by$. Let $c$ be a common divisor of $a$ and $b$. Then
$$\forall x', y' \in \mathbb{Z}: c | ax' + by'\Longrightarrow\exists q_1, q_2 \in \mathbb{Z}\,\,\, s.t.\,\,\, c q_1 = ax' + by'\,\,,\,\,cq_2 = g$$
So
$$gcd(a, b)=g=c q_2 = c q_1 \frac{q_2}{q_1} = \left(\frac{q_2}{q_1}x'\right)a + \left(\frac{q_2}{q_1}y'\right)b$$ So if we have $q_1|q_2$ then we found $\gcd(a,b)=ax'+by'$ for all $x', y' \in \mathbb{Z}$.
Now
$$\frac{q_1}{q_2}=\frac{c q_1}{c q_2} = \frac{ax'+by'}{g}$$ but $g|a$ and $g|b$ so $\exists q_3 \in \mathbb{Z}: \frac{q_1}{q_2}=q_3 \Rightarrow q_1=q_2 q_3 \Rightarrow q_1 | q_2\,$ . QED.
Best Answer
The nicest proof I know is as follows:
Consider the set $S = \{ax+by>0 : a,b \in \mathbb{Z}\}$. Let $d = \min S$. We now show the following:
If we can show those two things then it is trivial that $d$ is the greatest common divisor of $a$ and $b$, and therefore that the greatest common divisor of $a$ and $b$ is of the form $ax+by$.
To show that $d$ divides $a$ and $b$: suppose for a contradiction that $d$ does not divide $a$. Then $a=qd+r$, where $q\ge 0$ and $0<r<d$. Since $a=qd+r$, $qd=a-r$, and since we have that $d=ax+by$, $q(ax+by)=a-r$, so $r=a(1-qx)-bqy$. So $r$ is a linear combination of $a$ and $b$, and since $r>0$, that means that $r\in S$. Since $r<d$ and we had supposed $d$ to be $\min S$, we have a contradiction. So $d$ must divide $a$.
An identical argument proves that $d$ must divide $b$.
We now want to show that any common divisor of $a$ and $b$ must divide $d$. This is easy to show: if $a=uc$ and $b=vc$, then $d=ax+by=c(ux+vy)$, so $c$ divides $d$.
Therefore, $d$ is the greatest common divisor of $a$ and $b$, and is of the form $ax+by$.