Proof: By Euclid
(related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)
- For if [$A$ and $C$] are not prime (to one another) then some number will measure them.
- Let it measure (them), and let it be $D$.
- And $A$ is odd.
- Thus, $D$ (is) also odd.
- And since $D$, which is odd, measures $C$, and $C$ is even, [$D$] will thus also measure half of $C$ [Prop. 9.30].
- And $B$ is half of $C$.
- Thus, $D$ measures $B$.
- And it also measures $A$.
- Thus, $D$ measures (both) $A$ and $B$, (despite) them being prime to one another.
- The very thing is impossible.
- Thus, $A$ is not unprime to $C$.
- Thus, $A$ and $C$ are prime to one another.
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"