Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference)

(Proposition 8 from Book 7 of Euclid's “Elements”)

If a number is those parts of a number that a (part) taken away (is) of a (part) taken away then the remainder will also be the same parts of the remainder that the whole (is) of the whole.


Modern Formulation

See divisibility law no. 9.


This proposition states (for integers $0 < r_1 < r_0 < AL$ and $0 < m < n$) $$\begin{array}{rclclc}CD&=&AB+r_0&=&n\cdot AL+r_0&\wedge\\ CF&=&AE+r_1&=&m\cdot AL+r_1\\ &\Downarrow&\\ CD-CF&=&(AB-AE)+(r_0-r_1)&=&(n-m)AL+(r_0-r_1) \end{array}$$

with $0 < (r_0-r_1) < AL.$ In particular,

$$AL\not\mid (CD-CF)\Rightarrow AL\not\mid CD\vee AL\not\mid CF.$$

Proofs: 1

Proofs: 1

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