Proof: By Euclid
(related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
 For since $B$ is a part of $A$ called the same as $C$, and the unit $D$ is also a part of $C$ called the same as it (i.e., $D$ is the $C$th part of $C$), thus which(ever) part the unit $D$ is of the number $C$, $B$ is also the same part of $A$.
 Thus, the unit $D$ measures the number $C$ as many times as $B$ (measures) $A$.
 Thus, alternately, the unit $D$ measures the number $B$ as many times as $C$ (measures) $A$ [Prop. 7.15].
 Thus, $C$ measures $A$.
 (Which is) the very thing it was required to show.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"