Proof: By Euclid
(related to Proposition: 7.29: Prime not Divisor implies Co-prime)
- For if $B$ and $A$ are not prime to one another then some number will measure them.
- Let $C$ measure (them).
- Since $C$ measures $B$, and $A$ does not measure $B$, $C$ is thus not the same as $A$.
- And since $C$ measures $B$ and $A$, it thus also measures $A$, which is prime, (despite) not being the same as it.
- The very thing is impossible.
- Thus, some number cannot measure (both) $B$ and $A$.
- 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"
