Proof: By Euclid
(related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Coprime)
 For if they are not prime to one another then some number will measure them.
 Let it (so measure them), and let it be $C$.
 And as many times as $C$ measures $A$, so many units let there be in $D$.
 And as many times as $C$ measures $B$, so many units let there be in $E$.
 Since $C$ measures $A$ according to the units in $D$, $C$ has thus made $A$ (by) multiplying $D$ [Def. 7.15] .
 So, for the same (reasons), $C$ has also made $B$ (by) multiplying $E$.
 So the number $C$ has made $A$ and $B$ (by) multiplying the two numbers $D$ and $E$ (respectively).
 Thus, as $D$ is to $E$, so $A$ (is) to $B$ [Prop. 7.17].
 Thus, $D$ and $E$ are in the same ratio as $A$ and $B$, being less than them.
 The very thing is impossible.
 Thus, some number does not measure the numbers $A$ and $B$.
 Thus, $A$ and $B$ are prime to one another.
 (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"