Definition: 7.03: Proper Divisor

A number is part of a(nother) number, the lesser of the greater, when it measures the greater.

Modern Formulation

See proper divisor.

Notes

• Euclid requires a "part" to be a number strictly lesser then the number $$n > 1$$ it measures.
• Albeit some commentaries to Euclid's “Elements” state that the modern term of the "part" is the "divisor", this is not a correct interpretation, because Euclid never uses the term "part" for the trivial divisors $$\pm 1$$ and $$\pm n$$. Instead, in these cases he uses the notions "is measured by a unit" for $$1\mid n$$ or "is equal to" for $$n\mid n$$.
• Therefore, the modern term most close to the meaning of a "part" is "proper divisor".
• Please note that there is no definition in Euclid's “Elements” corresponding to the modern definition of a divisor.

Definitions: 1 2
Proofs: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23 24 25 26 27 28 29

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References

Bibliography

1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"