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.
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