Commentary on Proposition VII.24

Back to Proposition VII.24.

[Folio 0332 | p. 326]

If $a$, $b$ are both prime to $c$, then $ab$, $c$ are prime to one another. The proof is again by reductio ad absurdum. If $ab$, $c$ are not prime to one another, let them be measured by $d$ and be equal to $md$, $nd$, say, respectively.

Now, since $a$, $c$ are prime to one another and $d$ measures $c$,

$$a,\ d\text{ are prime to one another. \qquad [VII. 23]}$$

But, since $ab = md$,

$$d : a = b : m. \qquad\text{[VII. 19]}$$

Therefore [VII. 20] $d$ measures $b$, or $b = pd$, say. But $c = nd$.

Therefore $d$ measures both $b$ and $c$, which are therefore not prime to one another : which is impossible.

Therefore etc.