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:


 * $\gcd \set {a, b} = d \iff \dfrac a d, \dfrac b d \in \Z \land \gcd \set {\dfrac a d, \dfrac b d} = 1$

That is:


 * $\dfrac a {\gcd \set {a, b} } \perp \dfrac b {\gcd \set {a, b} }$

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

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

So:

Also presented as
It can be expressed so as not to include fractions:


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