Commentary on Proposition VII.28

Back to Proposition VII.28.

[Folio 0336 | p. 330]

If $a$, $b$ are prime to one another, $a + b$ will be prime to both $a$ and $b$ ; and conversely.

For suppose $(a + b)$, $a$ are not prime to one another. They must then have some common measure $d$.

Therefore $d$ also divides the difference $(a + b) - a$, or $b$, as well as $a$ ; and therefore $a$, $b$ are not prime to one another : which is contrary to the hypothesis.

$$\text{Therefore } a + b \text{ is prime to } a.$$

$$\text{Similarly } a + b \text{ is prime to } b.$$

The converse is proved in the same way.

Heiberg remarks on Euclid’s assumption that, if $c$ measures both $a$ and $b$, it also measures $a \pm b$. But it has already (VII. 1, 2) been assumed, more generally, as an axiom that, in the case supposed, $c$ measures $a \pm pb$.