Proposition: 7.05: Divisors Obey Distributive Law (Sum)

(Proposition 5 from Book 7 of Euclid's “Elements”)

If a number is part of a number, and another (number) is the same part of another, then the sum (of the leading numbers) will also be the same part of the sum (of the following numbers) that one (number) is of another. * For let a number $A$ be part of a [number] $BC$, and another (number) $D$ (be) the same part of another (number) $EF$ that $A$ (is) of $BC$. * I say that the sum $A$, $D$ is also the same part of the sum $BC$, $EF$ that $A$ (is) of $BC$.


Modern Formulation

See divisibility law no. 9.


This proposition states $$\begin{array}{rclc}BC&=&n\cdot A&\wedge\\ EF&=&n\cdot D\\ &\Downarrow&\\ BC+EF&=&n(A+D). \end{array}$$ In particular, $$n\mid BC\wedge n\mid EF\Rightarrow n\mid (BC+EF).$$

Proofs: 1

Proofs: 1 2 3 4 5 6

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016