Proposition 7.4 of Euclid's Elements Here is an online version of Book 7 of Elements where the following definitions and proposition can be found: https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html
Note: In his books on number theory, Euclid uses the word "measures" in place of "divides".
Definitions:

  
*
  
*A unit is that by virtue of which each of the things that exist is called one.
  
*A number is a multitude composed of units. (Thus, a number is a positive integer greater than one.)
  
*A number is a part of a(nother) number, the less of the greater, when it measures the greater;
  
*But (the lesser is) parts (of the greater) when it does not measure it.

Proposition 7.4 says:

Any number is either a part or parts of any (other) number, the less of the
  greater.

Euclid gives a somewhat long proof of this but isn't it obvious? If $a<b$, $a$ either measures $b$ or doesn't measure $b$. Thus, $a$ is either a part or parts of $b$.
What am I missing?

Euclid's proof isn't much longer than that.  He breaks things into three cases.  Assuming that $a<b$ are positive integers:


*

*If $a\mid b$, then $a$ is a part of $b$ (e.g. $2$ is a part of $4$)

*If $(a,b)=1$, then $a$ are parts of $b$ (e.g. $3$ are parts of $5$)

*If $(a,b)>1$ but $a\not\mid b$, then $a$ are parts of $b$ (e.g. $4$ are parts of $6$)


Yes, today we would merge those last two cases without a second thought.  But it seems from the commentary that there was a philosophical squabble at the time over whether $1$ was a proper number, so Euclid evidently felt it was worth a separate case.

Your question asked "What am I missing?"  TBH, I don't know.  Was this it?
