Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime
(Proposition 22 from Book 7 of Euclid's “Elements”)
The least numbers of those (numbers) having the same ratio as they are prime to one another.
* Let $A$ and $B$ be the least numbers of those (numbers) having the same ratio as them.
* I say that $A$ and $B$ are prime to one another.
This is the converse of Prop 7.21. If two natural numbers form a fraction $\frac AB$ and they are the smallest numbers forming this fraction, then they are co-prime.
Table of Contents
Proofs: 1 2 3 4
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016