Exercise 26, § 1

Let x, y be two coprime integers ; then every integer z such that $|z| < |xy|$ can be expressed, in either one or two ways, in the form $z = ux + vy$ where u and v are integers with $|u| < |y|$ and $|v| < |x|$. If z can be expressed in two different ways in this form, then neither x nor y can divide z ; the example $x = 5, y = 7, z = 12$ shows that the converse fails.