Integer Combination of Coprime Integers/Necessary Condition

Theorem
Let $a, b \in \Z$ be integers, not both zero.

Let $a$ and $b$ be such that there exists an integer combination of them equal to $1$.

Then $a$ and $b$ are coprime:


 * $\forall a, b \in \Z: \exists m, n \in \Z: m a + n b = 1 \implies a \perp b$

In such an integer combination $m a + n b = 1$, the integers $m$ and $n$ are also coprime.