# Integer Coprime to Factors is Coprime to Whole

## Theorem

Let $a, b, c \in \Z$ be integers.

Let:

- $a \perp b$
- $a \perp c$

where $\perp$ denotes coprimality.

Then:

- $a \perp b c$

In the words of Euclid:

(*The Elements*: Book $\text{VII}$: Proposition $24$)

### Integer Coprime to all Factors is Coprime to Whole

Let $a, b \in \Z$.

Let $\ds b = \prod_{j \mathop = 1}^r b_j$

Let $a$ be coprime to each of $b_1, \ldots, b_r$.

Then $a$ is coprime to $b$.

## Proof 1

*This proof follows the structure of Euclid's proof, if not its detail.*

Let $a, b, c \in \Z$ such that $c$ is coprime to each of $a$ and $b$.

Let $d = a b$.

Aiming for a contradiction, suppose $c$ and $d$ are not coprime.

Then:

- $\exists e \in \Z_{>1}: e \divides c, e \divides d$

We have that $c \perp a$ and $e \divides c$

From Proposition $23$ of Book $\text{VII} $: Divisor of One of Coprime Numbers is Coprime to Other:

- $e \perp a$

As $e \divides d$ it follows by definition of divisibility that:

- $\exists f \in \Z: e f = d$

We also have that:

- $a b = d$

So:

- $a b = e f$

But from Proposition $19$ of Book $\text{VII} $: Relation of Ratios to Products:

- $e : a = b : f$

or in more contemporary language:

- $\dfrac a e = \dfrac b f$

But $a \perp e$.

From Proposition $21$ of Book $\text{VII} $: Coprime Numbers form Fraction in Lowest Terms, $\dfrac a e$ is in canonical form.

From Proposition $20$ of Book $\text{VII} $: Ratios of Fractions in Lowest Terms:

- $e \divides b$

But we already have that:

- $e \divides c$

So $e$ is a divisor of both $b$ and $c$.

But this is a contradiction of the assertion that $c$ and $b$ are coprime.

Hence the result.

$\blacksquare$

## Proof 2

Let $a, b, c \in \Z$ such that $a$ is coprime to each of $b$ and $c$.

We have:

\(\ds a\) | \(\perp\) | \(\ds b\) | ||||||||||||

\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \exists x, y \in \Z: \, \) | \(\ds 1\) | \(=\) | \(\ds a x + b y\) | Integer Combination of Coprime Integers | ||||||||

\(\ds a\) | \(\perp\) | \(\ds c\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \exists u, v \in \Z: \, \) | \(\ds 1\) | \(=\) | \(\ds a u + c v\) | Integer Combination of Coprime Integers | |||||||||

\(\ds \) | \(=\) | \(\ds \paren {a x + b y} \paren {a u + c v}\) | from $(1)$ | |||||||||||

\(\ds \) | \(=\) | \(\ds a^2 u x + a c v x + a b y u + b c y v\) | multiplying out | |||||||||||

\(\ds \) | \(=\) | \(\ds a \paren {a u x + c v x + b y u} + b c \paren {y v}\) | rearranging | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds a\) | \(\perp\) | \(\ds b c\) | Integer Combination of Coprime Integers |

$\blacksquare$

## Historical Note

This proof is Proposition $24$ of Book $\text{VII}$ of Euclid's *The Elements*.