Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer

(Proposition 38 from Book 7 of Euclid's “Elements”)

If a number has any part whatever then it will be measured by a number called the same as the part. * For let the number $A$ have any part whatever, $B$. * And let the [number] $C$ be called the same as the part $B$ (i.e., $B$ is the $C$th part of $A$). * I say that $C$ measures $A$.


Modern Formulation

If $\frac AC=B$ for some natural number $B,$ then $A$ is a multiple of $C.$

Proofs: 1

Proofs: 1

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