Modulo Equivalence Classes 
Let $d$ be a fixed positive integer. Define a binary relation $\equiv_d$ on $\mathbb Z$ by $m\equiv_d n$ whenever $m-n$ is divisible by $d$. Determine all the equivalence classes under $\equiv_d$. How many are there? Verify that the equivalence classes form a partition of $\mathbb Z$. 

I am trying to figure this out and I am thinking there would be an infinite number of equivalence classes. Whatever $d$ is I can come up with an infinite combinations of $m-n$ is divisible by $d$. As for the Classes being a Partition of $\mathbb Z$, couldn't $m,n$ be non integers? Thanks in advance for any help.
 A: Equivalence Classes
Essential to understanding modulo classes is understand equivalence classes. 

For a relation $\sim$ on a set $A$, an equivalence class $[a]$ is defined as all elements $b\in A$ such that $a \sim b$. Using notation, $[a]:=\{b\in A: a\sim b\}$. 

There are a few important properties of equivalence classes: 


*

*If $a\sim b$, then $[a]=[b]$.

*If $a\not\sim b$, then $[a]\cap [b] = \emptyset$


You should attempt to prove these two yourself.
Modulo Classes
The relation you defined, $\equiv_d$ on $\mathbb{Z}$, is an equivalence class. To give you some insight into how these classes work, let's examine the simple case of $d=4$. Let's try and find $[0]$ and $[1]$. 
Plugging in $n=0$, we get that $m\equiv_4 0$ if $m-0$ is divisible by $4$. Next, plugging in $n=1$, we get that $m\equiv_4 1$ if $m - 1$ is divisible by $4$. So, we have that


*

*$[0] = \{4k:k\in\mathbb{Z}\}$.

*$[1] = \{4k + 1: k\in\mathbb{Z}\}$.


You should try the following:


*

*Find the rest of the equivalence classes. Do these form a partition of $\mathbb{Z}$? Why or why not?

*Prove that $\equiv_d$ is indeed an equivalence relation on $\mathbb{Z}$.

*Find the answers to your questions for the general case, after getting comfortable with various values of $d$.
