Divisor of One of Coprime Numbers is Coprime to Other/Proof 2

Theorem

 * If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.

Proof
Let $A, B$ be two numbers which are prime to one another.

Let $C$ be any number greater than $1$ which measures $A$.


 * Euclid-VII-23.png

Suppose $C$ and $B$ are not prime to one another.

Then some number $D$ will measure them both.

We have that $D$ measures $C$ and $C$ measures $A$.

So $D$ measures $A$.

But $D$ also measures $B$.

So $D$ measures $A$ and $B$ which are prime to one another.

By, this is a contradiction.

Therefore there can be no such $D$ that measures both $B$ and $C$.

That is, $B$ and $C$ are prime to one another.