Proposition: 7.27: Powers of Coprime Numbers are Coprime
(Proposition 27 from Book 7 of Euclid's “Elements”)
If two numbers are prime to one another and each makes some (number by) multiplying itself then the numbers created from them will be prime to one another, and if the original (numbers) make some (more numbers by) multiplying the created (numbers) then these will also be prime to one another [and this always happens with the extremes].
Modern Formulation
If $a$ is prime to $b$, then $a^2$ is also prime to $b^2$, as well as $a^3$ to $b^3$, etc., where all symbols denote numbers.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016