Proposition: 7.07: Divisors Obey Distributive Law (Difference)
(Proposition 7 from Book 7 of Euclid's “Elements”)
If a number is that part of a number that a (part) taken away (is) of a (part) taken away then the remainder will also be the same part of the remainder that the whole (is) of the whole.
- For let a number $AB$ be that part 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 part of the remainder $FD$ that the whole $AB$ (is) of the whole $CD$.
Modern Formulation
See divisibility law no. 6.
Notes
This proposition states
$$\begin{array}{rcl}
\underbrace{n\cdot (AE+EB)}_{n\cdot AB}=\underbrace{n\cdot AB}_{CD}&=&\underbrace{n\cdot AE}_{CF}+\underbrace{n\cdot EB}_{FD}\\
&\Downarrow&\\
n\cdot EB&=&\underbrace{n\cdot (AB-AE)}_{=FD}.
\end{array}$$
In particular,
$$n\mid (CF+FD)\wedge n\mid CF\Rightarrow n\mid FD.$$
Table of Contents
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Proofs: 1
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016
