◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--7-elementary-number-theory / Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole

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 $C.$

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3

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References

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", https://proofwiki.org/wiki/Main_Page, 2016