Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole

(Proposition 24 from Book 7 of Euclid's “Elements”)

If two numbers are prime to some number then the number created from (multiplying) the former (two numbers) will also be prime to the latter (number). * For let $A$ and $B$ be two numbers (which are both) prime to some number $C$. * And let $A$ make $D$ (by) multiplying $B$. * I say that $C$ and $D$ are prime to one another.


Modern Formulation

If $A$ and $B$ are both co-prime to $C,$ then also the product $AB$ is co-prime to $C.$

Proofs: 1

Proofs: 1 2 3

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