Elements of $\mathbb{Z}/(n)$ Let $(n) = \{ \lambda n | \lambda \in \mathbb{Z} \}$. In my book it has shown that every element in  $\mathbb{Z}/(n)$ can be expressed uniquely in the form $r + (n)$ where $0 \leq r \leq n-1$ now I look at a practical application of this, say:
$\mathbb{Z}/6 = \{ 0,1,2,3,4,5 \}$ and when computing elements in this set we just do it in modulo $6$, so everytime I see a $6$ I write $0$, so for $\mathbb{Z}/(n)$ everytime I see $n$ I write $0$. However, I don't see how this leads on from the theorem I stated above, i.e. if we know that every element in  $\mathbb{Z}/(n)$ can be represented in the form $r + (n)$ with $ 0 \leq r \leq n-1$ why does that mean, when I see $n$ I write $0$? I'm just not comfortable with $r + (n)$ I mean, what is the $(n)$ doing there?

Notice that $r$ is the remainder of the division of any integer number by $n$. And as $0 \leq r < n$ then you may choose a class representant to the residue class $[r]$ then 
$$[r] = r + (n) = \{r + x ; x \in (n)\}$$
You may also notice that 
$$a \equiv b\mod (n) \iff b-a \in (n)$$
