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.

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.

Proofs: 1

Proofs: 1 2 3 4

Thank you to the contributors under CC BY-SA 4.0!



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",, 2016