The supremum-like definition of greatest common divisor I have reasonable experience with analysis, but I have just recently begun studying abstract algebra from Dummit and Foote.  
I am frightened that I am already getting tripped up on pp.4, with the definition of greatest common divisor.
My intuitive understanding of greatest common divisor, from grade school, would be the maximal common element of two integers' factor trees.   I am having trouble aligning this with Dummit and Foote's definition, which parallels the definition of a supremum.  
First, they make a predefinition:  if $a,b \in \mathbb{Z}$ with $a \neq 0$, then they say $a$ divides $b$, or $a|b$, if $\exists c \in \mathbb{Z} : b= ac$.  
Then, they proceed with the definition: if $a,b \in \mathbb{Z} - \{0\}$, then there exists a unique positive integer $d$, called the greatest common divisor of $a$ and $b$, satisfying:


*

*$d | a$ and $d | b$ (so $d$ is a common divisor of $a$ and $b$)

*if $e|a$ and $e|b$ then $e|d$ (so $d$ is the greatest such divisor)


Now I can't actually figure out how to prove the second property.  
What I mean is that, suppose I assume $d|a$,$d|b$,$e|a$,$e|b$.  Then why does $d>e \implies e|d$? 

Let $g$ be the gcd of $a$ and $b$. The we can write
$$a=m*g$$
$$b=n*g$$
Where $m$ and $n$ are co-prime ie their gcd is 1. ( This is evident ). Now let us assume $e$ divides both $a$ and $b$. Then $e$ can't be a divisor of $m$ and $n$ because they are co-prime , it means $e$ has to be a divisor of $g$ so that it can divide both $a$ and $b$.
