Numbers forming Fraction in Lowest Terms are Coprime

Theorem

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

Proof
Let $A, B$ be (natural) numbers which are the least of those which have the same ratio with them.

We need to show that $A$ and $B$ are prime to one another.


 * Euclid-VII-22.png

Aiming for a contradiction, suppose $A$ and $B$ are not coprime.

Then by definition there exists some (natural) number $C > 1$ which measures them both.

As many times as $C$ measures $A$, let that many units be in $D$.

As many times as $C$ measures $B$, let that many units be in $E$.

So by :
 * $A = C \times D$
 * $B = C \times E$

Thus by :
 * $D : E = A : B$

So $D$ and $E$ are in the same ratio with $A$ and $B$.

But by Integer Absolute Value not less than Divisors: Corollary:
 * $D < A$
 * $E < B$

This contradicts our hypothesis that $A$ and $B$ are the least of those numbers that are in the same ratio with $A$ and $B$.

It follows that $A$ and $B$ must be prime to one another.