Question on quotient groups

Question

I know that $\mathbb{Z}/n\mathbb{Z}$ stands for the quotient group of integers mod $n$. To be a little more specific, we define the relation: $$a \equiv b \hspace{0.1cm} (\mbox{mod $n$}) \iff n\mid (a-b)$$ and this is an equivalence relation. Thus, $\mathbb{Z}/n\mathbb{Z}$ is defined to be the set of all equivalence classes of this relation. We can prove that: $$\mathbb{Z}/n\mathbb{Z} = \{[0],...,[n-1]\}$$ This being said, I would like to understand the meaning of $\mathbb{Z}^{d}/n\mathbb{Z}^{d}$ and $\mathbb{R}^{d}/n\mathbb{Z}^{d}$. I'm having trouble understanding these objects because I don't know how to define 'divisibility' in $\mathbb{Z}^{d}$ and $\mathbb{R}^{d}$. Do we need to demand componentwise divisibility? How to define these groups?

Answer 1

An element of $n\mathbb Z^d$ is $n$ times an element of $\mathbb Z^d$. So it's a list of $d$ integers, all of which are divisible by $n$. Two elements of $\mathbb Z^d$ are equivalent in $\mathbb Z^d/n\mathbb Z^d$ if and only if their difference is in $n\mathbb Z^d$. So $(p_0,\dots,p_{d-1})=(q_0,\dots,q_{d-1})$ if and only if $n|(p_i-q_i)$ for all $i<d$.