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.
If $A$ and $B$ are both co-prime to $C,$ then also the product $AB$ is co-prime to $C.$
Table of Contents
Proofs: 1 2 3
Thank you to the contributors under CC BY-SA 4.0!
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016