Factoring vs dividing by $\mathbb Z$

Question

Is $x|8$ the same as $x \equiv 8 / \mathbb Z$ when $x \in \mathbb Z$?

Answer 1

No. $x\mid 8$ is the proposition "$x$ divides $8$", wich is defined as $\exists m\in\mathbb Z : mx = 8$. This can be either true or false.
$8/\mathbb Z$ on the other hand is not even proper notation.

Whats similar to your question and still true is $$x\mid 8 \Leftrightarrow \color{maroon}{8 \in x\mathbb Z} = \{xm : m\in\mathbb Z\}$$ Note the "element of" ($\in$) versus the equivalent to ($\equiv$).

Another somehow reasonable equivalent statement would be $$x\in 8/\mathbb Z^*$$ where $\mathbb Z^* = \mathbb Z \setminus \{0\}$ and $k/A := \{k/a : a \in A\}$ would have to be defined.