Proposition: 8.06: First Element of Geometric Progression not dividing Second
(Proposition 6 from Book 8 of Euclid's “Elements”)
If there are any multitude whatsoever of numbers in continued proportion, and the first does not measure the second, then no other (number) will measure any other (number) either.
 Let $A$, $B$, $C$, $D$, $E$ be any multitude whatsoever of numbers in continued proportion, and let $A$ not measure $B$.
 I say that no other (number) will measure any other (number) either.
Modern Formulation
If $A,B,C,D,E$ are given with $B=An,$ $C=An^2,$ $D=An^3,$ $E=An^4,$ and $n$ is not a positive integer, then $A$ is not a divisor of $B.$ In this geometric progression, none of the numbers will be a multiple of any of the proceeding numbers.
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016