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.


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

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