Proof: By Euclid
(related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
- For if $C$ and $B$ are not prime to one another then [some] number will measure $C$ and $B$.
- Let it (so) measure (them), and let it be $D$.
- Since $D$ measures $C$, and $C$ measures $A$, $D$ thus also measures $A$.
- And ($D$) also measures $B$.
- Thus, $D$ measures $A$ and $B$, which are prime to one another.
- The very thing is impossible.
- Thus, some number does not measure the numbers $C$ and $B$.
- Thus, $C$ and $B$ are prime to one another.
- (Which is) the very thing it was required to show.
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Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"