Definition: 7.05: Multiple, Number Multiplying another Number
(Definition 5 and 15 from Book 7 of Euclid's “Elements”)
And the greater (number is) a multiple of the lesser (number) when it is measured by the lesser.
- Euclid requires a multiple to be strictly greater than any of its divisors.
- Note that the modern definition of the multiple does not make this restriction.
Definitions: 1 2 3 4
Proofs: 5 6 7 8 9 10
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"