Proposition: 7.39: Least Number with Three Given Fractions

(Proposition 39 from Book 7 of Euclid's “Elements”)

To find the least number that will have given parts. * Let $A$, $B$, and $C$ be the given parts. * So it is required to find the least number which will have the parts $A$, $B$, and $C$ (i.e., an $A$th part, a $B$th part, and a $C$th part).


Modern Formulation

If $A=\frac 1D,$ $B=\frac 1E,$ $C=\frac 1F,$ and $G=\operatorname{lcm}(D,E,F)$ is the least common multiple of these numbers with $D=\frac G{d},$ $E=\frac G{e},$ $F=\frac G{f},$ then $A=d/G,$ $B=e/G,$ and $C=f/G.$

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