# Multiples of Divisors obey Distributive Law

## Theorem

In the words of Euclid:

*If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.*

(*The Elements*: Book $\text{VII}$: Proposition $6$)

In modern algebraic language:

- $a = \dfrac m n b, c = \dfrac m n d \implies a + c = \dfrac m n \paren {b + d}$

## Euclid's Proof

Let the (natural) number $AB$ be an aliquant part of the (natural) number $C$.

Let the (natural) number $DE$ be the same aliquant part of another (natural) number $F$ that $AB$ is of $C$.

We need to show that $AB + DE$ is the same aliquant part of $C + F$.

We have that whatever aliquant part $AB$ is of $C$, $DE$ is also the same aliquant part as $F$.

It follows that as many aliquant parts of $C$ as there are in $AB$, so many aliquant parts of $F$ are there also in $DE$.

Let $AB$ be divided into the aliquant parts of $C$, namely $AG, GB$, and $DE$ into the aliquant parts of $F$, namely $DH, HE$.

Thus the multitude of $AG, GB$ will be equal to the multitude of $DH, HE$.

We have that whatever aliquot part $AG$ is of $C$, the same aliquot part is $DH$ of $F$ also.

Therefore, from Proposition $5$ of Book $\text{VII} $: Divisors obey Distributive Law, whatever aliquot part $AG$ is of $C$, the same aliquot part also is $AG + DH$ of $C + F$ also.

For the same reason, whatever aliquot part $GB$ is of $C$, the same aliquot part also is $GB + HE$ of $C + F$.

Therefore whatever aliquant part $AB$ is of $C$, the same aliquant part also is $AB + DE$ of the $C + F$

$\blacksquare$

## Modern Proof

A direct application of the Distributive Property:

- $\dfrac m n b + \dfrac m n d = \dfrac m n \paren {b + d}$

$\blacksquare$

## Historical Note

This proof is Proposition $6$ of Book $\text{VII}$ of Euclid's *The Elements*.