Commentary on Proposition VII.26

Back to Proposition VII.26.

[Folio 0333 | p. 327]

If both $a$ and $b$ are prime to each of two numbers $c$, $d$, then $ab$, $cd$ will be prime to one another.

Since $a$, $b$ are both prime to $c$,

$$ab,\ c\text{ are prime to one another. \qquad [VII. 24]}$$

Similarly

$$ab,\ d\text{ are prime to one another.}$$

Therefore $c$, $d$ are both prime to $ab$,

$$\text{and so therefore is }cd.\text{ \qquad [VII. 24]}$$