Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other

(Proposition 23 from Book 7 of Euclid's “Elements”)

If two numbers are prime to one another then a number measuring one of them will be prime to the remaining (one). * Let $A$ and $B$ be two numbers (which are) prime to one another, and let some number $C$ measure $A$. * I say that $C$ and $B$ are also prime to one another.


Modern Formulation

If $A$ and $B$ are co-prime and $C$ is a divisor of $A$, then $C$ and $B$ are also co-prime.

Proofs: 1

Proofs: 1

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