Proposition: 7.30: Euclidean Lemma

(Proposition 30 from Book 7 of Euclid's “Elements”)

If two numbers make some (number by) multiplying one another, and some prime number measures the number (so) created from them, then it will also measure one of the original (numbers). * For let two numbers $A$ and $B$ make $C$ (by) multiplying one another, and let some prime number $D$ measure $C$. * I say that $D$ measures one of $A$ and $B$.


Modern Formulation

see generalized Euclidean lemma.

Proofs: 1

Proofs: 1

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




  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

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