I don't think someone has posted this question yet. One looks similar, but I wasn't sure. Sorry in advance if it is.
I think I am making things up toward the end. I wanted to make a squeeze argument.
Suppose we are given:
$ax \equiv 1 \pmod{y}$ and $by \equiv 1 \pmod{x}$
where $a,b,x,y \in \mathbb{Z} $
Then it is true that,
$ \begin{align} y &| 1-ax \\ x &| 1-by \end{align}$ And since, $\gcd(y,x) | x,y$, it follows that the $\gcd(x,y) | 1-ax \ $ and $1-by$. So suppose the $\gcd(x,y) \geq 1$, then $ \begin{align} 1 \leq \gcd(x,y) \ | \ x \ | \ 1 -by \end{align}$
Similarly,
$ \begin{align} 1 \leq \gcd(x,y) \ | \ y \ | \ 1 -ax \end{align}$
I wanted to say these last parts were less than or equal to one. But now I don't think that's true.