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Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers

Euclid's Formulation

Any number is either part or parts of any (other) number, the lesser of the greater. * Let $A$ and $BC$ be two numbers, and let $BC$ be the lesser. * I say that $BC$ is either part or parts of $A$.

Modern Formulation

For two positive integers $0 < a < b$ we have either $a\mid b\wedge a\neq b$ or $a\not\mid d.$

Notes

• In the modern formulation, this is part of the definition of divisors and not a separate theorem.

Table of Contents

Proofs: 1

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Proofs: 1

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References

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", https://proofwiki.org/wiki/Main_Page, 2016