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For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number ofsolutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residuesmodulus $c$ is calculated. We establish a direct relation with those modularsolutions and the distances between points of a modular hyperbola. Furthermore,for certain composite moduli $c$, an asymptotic formula for quotients betweenthe number of solutions and $c$ is provided. Finally, an algorithm for integerfactorization using such solutions is presented.

Canonical form of modular hyperbolas with an application to integer factorization