Divisors obey Distributive Law/Proof 2

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{VII}$: Proposition $5$)

In modern algebraic language:

$a = \dfrac 1 n b, c = \dfrac 1 n d \implies a + c = \dfrac 1 n \paren {b + d}$

Proof

A direct application of the Distributive Property:

$\dfrac 1 n b + \dfrac 1 n d = \dfrac 1 n \paren {b + d}$

$\blacksquare$