Computing Factor Group I am reading John Fraleigh's First Course in Abstract Algebra, $\S$36 on the Second Isomorphism Theorem which says that if $H < G$ and $N \triangleleft G$, then
$$(HN)/N \cong H/(H \cap N).$$
He then goes on with an example which I delineate for easy reference:


*

*Let $G = \mathbb Z \times \mathbb Z \times \mathbb Z,$ and subgroup $H = \mathbb Z \times \mathbb Z \times \{0\},$ and normal subgroup $N = \{0\} \times \mathbb Z \times \mathbb Z.$   

*Then clearly $HN = \mathbb Z \times \mathbb Z \times \mathbb Z$ and $H \cap N = \{0\} \times \mathbb Z \times \{0\}.$  

*We have $(HN)/N \cong \mathbb Z$ and we also have $H/(H \cap N) \cong \mathbb Z.$
I am comfortable with (1) and (2), I am familiar with definition of factor group but how do you break down that $(HN)/N$ is isomorphic to $\mathbb Z$ and $H/(H \cap N)$ is isomorphic to $ \mathbb Z$ in (3), please?
Thank you for your time and effort.
 A: In, (HN)/N, anything in the HN of the form (a,x,y) is going to end up in same equivalence class (here 'a' is fixed whereas x and y can vary). You can verify that using the definition of equivalence classes modulo N. 
This shows that the equivalence classes modulo N, i.e. (HN)/N is determined by the first element in the 3-tuple of HN. The isomorphism between (HN)/N is trivial where each element (a,b,c) is sent to 'a' in $\mathbb Z$.
Similarly, in case of  H/(H $\cap N$)$\cong$$\mathbb Z$, every element of the form (a,x,0) is sent to the same equivalence class (where 'a' is fixed and x may vary). Hence, there is a clear isomorphism between H/(H $\cap N$)$\cong$$\mathbb Z$ where each element (a,x,y) is sent to 'a' in $\mathbb Z$. 
I hope that helps
A: It is a general fact that, given groups $G_1,\ldots,G_n$ and normal subgroups $H_i\trianglelefteq G_i$,
$$(H_1\times\cdots \times H_n)\trianglelefteq(G_1\times\cdots\times G_n)$$
and that
$$(G_1\times\cdots\times G_n)/(H_1\times\cdots\times H_n)\cong (G_1/H_1)\times\cdots\times (G_n/H_n)$$
For example, in your situation,
$$\begin{align*}
(HN)/N&=G/N\\
&=(\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z})/(\{0\}\times\mathbb{Z}\times\mathbb{Z})\\
&\cong(\mathbb{Z}/\{0\})\times(\mathbb{Z}/\mathbb{Z})\times(\mathbb{Z}/\mathbb{Z})\\
&\cong\mathbb{Z}\times(\text{trivial group})\times(\text{trivial group})\\
&\cong\mathbb{Z}
\end{align*}$$
