# 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. * For let the two numbers, $AB$ and $BC$, (which are) prime to one another, be laid down together. * I say that their sum $AC$ is also prime to each of $AB$ and $BC$. * Conversely, if $AC$ and $AB$ are prime to one another, then $AB$ and $BC$ are 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

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### References

#### 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", https://proofwiki.org/wiki/Main_Page, 2016