Proposition: 7.35: Least Common Multiple Divides Common Multiple

(Proposition 35 from Book 7 of Euclid's “Elements”)

If two numbers (both) measure some number then the least (number) measured by them will also measure the same (number). * For let two numbers, $A$ and $B$, (both) measure some number $CD$, and (let) $E$ (be the) least (number measured by both $A$ and $B$). * I say that $E$ also measures $CD$.


Modern Formulation

see least common multiple.

Proofs: 1

Proofs: 1 2

Thank you to the contributors under CC BY-SA 4.0!



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",, 2016