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.
 For let a number $AB$ be those parts of a number $CD$ that a (part) taken away $AE$ (is) of a (part) taken away $CF$.
 I say that the remainder $EB$ is also the same parts of the remainder $FD$ that the whole $AB$ (is) of the whole $CD$.
Modern Formulation
See divisibility law no. 9.
Notes
This proposition states (for integers $0 < r_1 < r_0 < AL$ and $0 < m < n$)
$$\begin{array}{rclclc}CD&=&AB+r_0&=&n\cdot AL+r_0&\wedge\\
CF&=&AE+r_1&=&m\cdot AL+r_1\\
&\Downarrow&\\
CDCF&=&(ABAE)+(r_0r_1)&=&(nm)AL+(r_0r_1)
\end{array}$$
with $0 < (r_0r_1) < AL.$ In particular,
$$AL\not\mid (CDCF)\Rightarrow AL\not\mid CD\vee AL\not\mid CF.$$
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016