Proposition: 7.27: Powers of Co-prime Numbers are Co-prime

(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]. * Let $A$ and $B$ be two numbers prime to one another, and let $A$ make $C$ (by) multiplying itself, and let it make $D$ (by) multiplying $C$. * And let $B$ make $E$ (by) multiplying itself, and let it make $F$ by multiplying $E$. * I say that $C$ and $E$, and $D$ and $F$, are prime to one another.


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.

Proofs: 1

Proofs: 1 2

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