Commentary on Proposition VII.30

Back to Proposition VII.30.

[Folio 0338 | p. 332]

If $c$, a prime number, measure $ab$, $c$ will measure either $a$ or $b$. Suppose $c$ does not measure $a$. Therefore $c$, $a$ are prime to one another. [VII. 29] Suppose

$$ab = mc.$$

Therefore

$$c : a = b : m. \quad \text{[VII. 19]}$$

Hence [VII. 20, 21] $c$ measures $b$. Similarly, if $c$ does not measure $b$, it measures $a$. Therefore it measures one or other of the two numbers $a$, $b$.