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:


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

That is:


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

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

So:

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