TITLE: What does it mean to say "a divides b"
QUESTION [11 upvotes]: I am not a number theorist and I am learning about relations.
I encountered a relation that says 

$a \leq b$ if $a$ divides $b$

Can someone clarify what it means to a number to divide another number?
Does it mean what I think? $a$ divides $b$ if $a | b \in \mathbb{Z}$?

So given a set $S = \{a, a^2, a^3, \ldots\}$, with relation $a | b \leftrightarrow a \leq b$, does the relation hold
  going from left to right or right to left? i.e. $a|a^2, a^2|a^3,
\ldots$

REPLY [9 votes]: We say $a$ divides $b$, denoted by $a | b$,  if $b$ is a multiple of $a$ (ie, $b$ is an integer multiple of $a$). Equivalently, $a |b$ iff $b=ka$ for some integer $k$.  
To remember what "$2$ divides $6$" means, perhaps you can remember the phrase "$2$ divides $6$ into $3$ parts".  Hence, $2 | 6$.   
Note that $2 | 0$ because $0$ is an integer multiple of $2$: $0 = k2$ for some integer $k$. Just take $k=0$.