For $a,b \in \Bbb Z$, if $ax+by=2$ for some $x$ and $y$ in $\Bbb Z$, then $(a,b) = 2$

Question

I understand a counterexample is when $a=4, b=-3$ and $x=y=2$.

Yet, I get confused because according to this Bezout's identity it states that $ax+by=(a,b)$ Therefore shouldn't the above statement be true or am I misunderstanding the identity.

Answer 1

$d=(a,b)$ is the smallest natural number for which there exist $x,y\in\mathbb{Z}$ such that $ax+by=d$, but it is not the only natural number that has this property. In fact, all multiples of $d$ will. So if $d=1$, as in your example, then all natural numbers have this property.

Answer 2

$ax+by=c$ only means $c$ is a multiple of $\gcd(a,b)$.