Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum)

(Proposition 6 from Book 7 of Euclid's “Elements”)

If a number is parts of a number, and another (number) is the same parts of another, then the sum (of the leading numbers) will also be the same parts of the sum (of the following numbers) that one (number) is of another.


Modern Formulation

See divisibility law no. 9.


This proposition states (for integers $0 < r_0 < AG$ and $0 < r_1 < DH$) $$\begin{array}{rclclc}C&=&AB+r_0&=&n\cdot AG+r_0&\wedge\\ F&=&DE+r_1&=&n\cdot DH+r_1\\ &\Downarrow&\\ C+F&=&(AB+DE)+(r_0+r_1)&=&n(AG+DH)+(r_0+r_1) \end{array}$$

with $0 < (r_0+r_1) < (AG+DH).$ In particular,

$$n\not\mid (C+F)\Rightarrow n\not\mid C\vee n\not\mid F.$$

Proofs: 1

Proofs: 1 2 3

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