Impossible pairs?

Relevant wiki: Bezout's Identity

Bézout's Identity states that for any pair of integers $m$ and $n$ , there exist integers $a$ and $b$ such that $am+bn = \mathrm{gcd}(m,n)$ . This is a special case.

The proof of Bézout's Identity involves considering the smallest integer $d$ of the form $am+bn$ . First, using the division algorithm, show that $d$ is a divisor of both $m$ and $n$ . Then, use the fact that $d=am+bn$ to argue that it is the greatest common divisor.