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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.

Modern Formulation

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

Mentioned in:

Proofs: 1 2 3 4

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016