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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.

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    Just a passing comment: more generally, let $a,b$ be elements of a commutative ring $R$ such that $ax \equiv 1 \pmod{y}$. Then $x$ is a unit in $R/(y)$, so $(x) + (y)$ is the unit ideal of $R/(y)$ and thus $(x)+(y) = R$. In other words, in general either one of the two conditions implies that $x$ and $y$ are **comaximal**.2012-07-10
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    Above, I should have said "let $a,x,y$ be elements...".2012-07-10

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