Coprime Numbers form Fraction in Lowest Terms

Theorem

In the words of Euclid:

Numbers prime to one another are the least of those which have the same ratio with them.

(The Elements: Book $\text{VII}$: 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 Proposition $20$ of Book $\text{VII} $: 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 Proposition $20$ of Book $\text{VII} $: 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 proof is Proposition $21$ of Book $\text{VII}$ of Euclid's The Elements.
It is the converse of Proposition $22$: Numbers forming Fraction in Lowest Terms are Coprime.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions