Proposition: 7.26: Product of Co-prime Pairs is Co-prime

(Proposition 26 from Book 7 of Euclid's “Elements”)

If two numbers are both prime to each of two numbers then the (numbers) created from (multiplying) them will also be prime to one another. * For let two numbers, $A$ and $B$, both be prime to each of two numbers, $C$ and $D$. * And let $A$ make $E$ (by) multiplying $B$, and let $C$ make $F$ (by) multiplying $D$. * I say that $E$ and $F$ are prime to one another.


Modern Formulation

Let $A$ and $C,$ $B$ and $C,$ $A$ and $D,$ $B$ and $D$ be co-prime. Then $AB$ is co-prime to $CD.$

Proofs: 1

Proofs: 1

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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