Proposition: 7.25: Square of Co-prime Number is Co-prime

Euclid's Formulation

If two numbers are prime to one another then the number created from (squaring) one of them will be prime to the remaining (number). * Let $A$ and $B$ be two numbers (which are) prime to one another. * And let $A$ make $C$ (by) multiplying itself. * I say that $B$ and $C$ are prime to one another.


Modern Formulation

If $A$ and $B$ are co-prime, then $A^2$ and $B$ are co-prime.

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