Integers Divided by GCD are Coprime/Proof 2

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:

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

if and only if:

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

that is:

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

where:

$\gcd$ denotes greatest common divisor

$\perp$ denotes coprimality.

Proof

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

We have:

$(1): d \divides a \iff \exists s \in \Z: a = d s$

$(2): d \divides b \iff \exists t \in \Z: b = d t$

We have to prove:

$\gcd \set {s, t} = 1$

Aiming for a contradiction, suppose $\gcd \set {s, t} \ne 1$.

So:

$(3): \exists k \in \N \setminus \set 1$ such that $k \divides s \land k \divides t$

So:

$(4): \exists m, n \in \N: s = k m, t = k n$

Substituting from $(4)$ in $(1)$ and $(2)$:

$a = d k m$, $b = d k n$

Therefore:

$ d k \divides a \land d k \divides b$

From $(3)$ we have:

\(\ds \)

\(\)

\(\ds k \in \N \land k \ne 1\)

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds k > 1\)

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds d k > d\)

As $d k$ is a common divisor of $a$ and $b$ greater than $d$, this contradicts $d = \gcd \set {a, b}$.

So our initial assumption that $\gcd \set {s, t} \ne 1$ is false.

Therefore, from Proof by Contradiction, we have:

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

$\blacksquare$