Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both

(Proposition 28 from Book 7 of Euclid's “Elements”)

If two numbers are prime to one another then their sum will also be prime to each of them. And if the sum (of two numbers) is prime to any one of them then the original numbers will also be prime to one another.


Modern Formulation

If $a$ and $b$ are co-prime, then $a$ and $a+b$ as well $b$ and $a+b$ are co-prime, and vice versa.

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