Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms

(Proposition 1 from Book 8 of Euclid's “Elements”)

If there are any multitude whatsoever of numbers in continued proportion, and the outermost of them are prime to one another, then the (numbers) are the least of those (numbers) having the same ratio as them.


Modern Formulation

This proposition states that if \(a,p,q\) are numbers (i.e. a positive integers) and \(p/q\) is a common ratio of the geometric progression. \[a,a\frac pq,a \left(\frac pq\right)^2,\ldots, a \left(\frac pq\right)^n\quad\quad( * )\]

for some \(n\) such that \(a \left(\frac pq\right)^n\) is also number (i.e. a positive integer), which is co-prime to \(a\), i.e. \[\gcd\left(a,a \left(\frac pq\right)^n\right)=1,\] then the numbers in the geometric progression \( ( * ) \) are the lowest possible with these properties (i.e. having the same common ratio \(p/q\) and having \(a \left(\frac pq\right)^n\) and \(a\) co-prime to each other).

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