Divisor of One of Coprime Numbers is Coprime to Other/Proof 2
Theorem
Let $a, b \in \N$ be numbers such that $a$ and $b$ are coprime:
- $a \perp b$
Let $c > 1$ be a divisor of $a$:
- $c \divides a$
Then $c$ and $b$ are coprime:
- $c \perp b$
In the words of Euclid:
- If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.
(The Elements: Book $\text{VII}$: Proposition $23$)
Proof
Let $A, B$ be two numbers which are prime to one another.
Let $C$ be any number greater than $1$ which measures $A$.
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 Book $\text{VII}$ Definition $12$: Relatively Prime, 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.
$\blacksquare$
Historical Note
This proof is Proposition $23$ of Book $\text{VII}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions