Notation question: $\mid $

Question

I have the following problem:

In this problem all variables range over $Z$, the set of all integers.

a. Prove that if $a \mid b$ and $a \mid c$, then $a\mid (b + c)$.

b. Prove that if $ac \mid bc$, and $c \neq 0$, then $a \mid b$.

This question has basically been asked before here. I just literally do not know what the vertical bar "$\mid$" means and somehow I can't find this information anywhere.

The best solution I can find is that it means "nand" in the context of boolean logic, e.g. $P\mid Q$ means $P$ and $Q$ are not both true. However this doesn't seem particularly relevant for integers.

Anything helps, thank you!

Answer 1

It means "divides".

$a \mid b$ iff there is $c$ such that $a \times c = b$.

Answer 2

It’s important to realize that in the sentence “$a\mid b$”, the vertical bar is the verb. Thus this sentence “$a|b$” is either true or false, depending on whether there is an integer $m$ for which $b=ma$. There is no numerical value, just, if you like, a boolean value. Some true sentences are $3\mid 12$, $3\mid 0$, and $0\mid 0$. Some false sentences are $3\mid 5$ and $0\mid 5$.