Integers Divided by GCD are Coprime

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

Let $d$ be a common divisor of $a$ and $b$, that is:
 * $\dfrac a d, \dfrac b d \in \Z$

Then:
 * $\gcd \set {a, b} = d$


 * $\gcd \set {\dfrac a d, \dfrac b d} = 1$
 * $\gcd \set {\dfrac a d, \dfrac b d} = 1$

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

where:
 * $\gcd$ denotes greatest common divisor
 * $\perp$ denotes coprimality.

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


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