Integers Divided by GCD are Coprime/Proof 3

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

Let $d$ be a common divisor of $a$ and $b$, that is:

$\dfrac a d, \dfrac b d \in \Z$


$\gcd \set {a, b} = d$

if and only if:

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

that is:

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


$\gcd$ denotes greatest common divisor
$\perp$ denotes coprimality.


Because $d$ is a common divisor of $a$ and $b$, we may form the expressions:

$a = d r$
$b = d s$

where $r, s \in \Z$.


\(\ds d\) \(=\) \(\ds \gcd \set {a, b}\) by hypothesis
\(\ds \) \(=\) \(\ds \gcd \set {d r, d s}\)
\(\ds \) \(=\) \(\ds d \gcd \set {r, s}\) GCD of Integers with Common Divisor
\(\ds \leadstoandfrom \ \ \) \(\ds 1\) \(=\) \(\ds \gcd \set {r, s}\) dividing through by $d$
\(\ds \) \(=\) \(\ds \gcd \set {\dfrac a d, \dfrac b d}\) Definition of $r$ and $s$