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Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms

Euclid's Formulation

Numbers prime to one another are the least of those (numbers) having the same ratio as them.

fig21e

Modern Formulation

If \frac{A}{B} is a reduced fraction, i.e. A and B are co-prime, then from all pairs of numbers having the same ration, A and B are the smallest.

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11
Propositions: 12


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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