# What does it mean to say "a divides b"

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. $a|a^2, a^2|a^3, \ldots$

• Let $(G,\circ)$ be a group. We then say that $a\mid b$ (read as "$a$ divides $b$") if, $$\exists c\in G\mid b=a\circ c$$
– user170039
Jun 24, 2016 at 4:31
• A common mistake is to confuse the statement $a \mid b$, which may be either true or false but is nonetheless a relationship between two numbers, with the quotient $a/b$ (i.e., a number). So "$a/b \in \Bbb Z$" makes sense, and may be true or false. But "$a \mid b \in \Bbb Z$" is not a well-formed statement. Jun 24, 2016 at 4:37
• "a divides b" means a and b are integers and there is an integer n, such that n x a = b; or, if you prefer $b/a \in \mathbb Z$, or if you prefer "a divides into b evenly with no remainder". The notation $a | b$ doesn't mean what you think it does. "|" isn't an operation that give a third value. $a | b$ is shorthand for the sentence "a divides b". So it goes left to right $a|a^2$ and $a^k | a^{k+m}$ etc. Jun 24, 2016 at 4:49
• @pjs36 Not that $a | b \iff b/a \in \mathbb Z$. and not the other way. Jun 24, 2016 at 4:51
• If the terminology is more familiar, "$a$ divides $b$" is the same as "$b$ is divisible by $a$".
– Mike
Jun 24, 2016 at 6:46

Given two integers $a$ and $b$, we say $a$ divides $b$ if there is an integer $c$ such that $b=ac$.
This is what $a$ divides $b$ means. The shorthand notation is $$a|b$$.
In your example, $$a|a^2\iff a\leq a^2$$ since by definition there exists $c$ such that $a^2 = ac$, namely $a = c$.
• Yikes, shouldn't it be $b$ divides $a$, since "b divison operator a = b/a = c$Jun 24, 2016 at 4:33 • No, you are reading the definition incorrectly. – Em. Jun 24, 2016 at 4:37 • Wait, were you calling the pipe symbol$|$the division operator? I'm not even sure what that means, but no$a\mid b$is not the same as$a/b$.$a/b$may not be an integer, like$3/4$, while$a\mid b$guarantees all intgers$b = ac$, like$4\mid 8$implies$8 = 4(2)$. – Em. Jun 24, 2016 at 4:55 • The pipe symbol "|" and the quotient symbol "/" are completely different things. That's the source of your confusion. "a|b" is shorthand for "a divides evenly into b with no remainder" whereas "a/b" is the result you get when you divide b into a. Different concepts. Note:$a | b \iff b/a \in \mathbb Z$. Jun 24, 2016 at 5:03 • @Niing Yes,$\frac{b}{a}$must be an integer. Hence,$a\neq 0$for$\frac{b}{a}$. For example, take a look at this question regarding$0|0$and why it can make sense. – Em. Mar 18, 2018 at 9:23 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\$.