Proposition: 7.28: Numbers are Coprime iff Sum is Coprime 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 coprime, then $a$ and $a+b$ as well $b$ and $a+b$ are coprime, and vice versa.
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