Integers Divided by GCD are Coprime

Theorem
Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their GCD:


 * $\displaystyle \gcd \left\{{a, b}\right\} = d \iff \frac a d, \frac b d \in \Z \land \gcd \left\{{\frac a d, \frac b d}\right\} = 1$

That is:


 * $\displaystyle \frac a {\gcd \left\{{a, b}\right\}} \perp \frac b {\gcd \left\{{a, b}\right\}}$

Alternatively it can be expressed so as not to include fractions:


 * $\gcd \left\{{a, b}\right\} = d \iff \exists s, t \in \Z: a = d s \land b = d t \land \gcd \left\{{s, t}\right\} = 1$

Proof
Let $d = \gcd \left\{{a, b}\right\}$.

We have:
 * $d \mathop \backslash a \iff \exists s \in \Z: a = d s$
 * $d \mathop \backslash b \iff \exists t \in \Z: b = d t$

So: