Numbers in Fractions in Lowest Terms are Coprime

Theorem

As Euclid defined it:

(Natural) numbers prime to one another are the least of those which have the same ratio with them.

(The Elements: Book VII: Proposition $21$)

Proof

Let $A, B$ be (natural) numbers which are prime to one another.

We need to show that $A$ and $B$ are the least of those which have the same ratio with them.

Suppose this were not the case.

Then there would be some numbers $C, D$ which are less than $A, B$ such that $A : B = C : D$.

From Book VII Proposition 20: Ratios of Fractions in Lowest Terms we have that $C$ measures $A$ the same number of times that $D$ measures $B$.

Now, as many times as $C$ measures $A$, let there be so many units in $E$.

Then from Book VII Proposition 20: Natural Number Multiplication is Commutative $E$ also measures $A$ according to the units in $C$.

For the same reason $E$ also measures $B$ according to the units in $D$.

Therefore $E$ measures $A$ and $B$, which are prime to one another, which is contrary to our initial hypothesis.

Hence the result.

$\blacksquare$

Historical Note

This is Proposition 21 of Book VII of Euclid's The Elements.