Equivalence Classes for 7 divides (x-y)

Question

How do I find the distinct equivalence classes for the relation $(x,y)\in R$ if and only if $7$ divides $(x-y)$?

Answer 1

So two numbers are related if $7$ divides the difference between them.

Let's say you have some $n\in \mathbb Z$.

$n$ might be expressed as $7k$ or $7k+1$ or $7k+2$ and so on for some $k \in \mathbb Z$. If $n=7k+7$ then let $l=k+1$ and instead say that $n=7l$.

In other words, every integer $n$ can be expressed as: $$n=7k + a$$$$ \text{where }\quad k \in \mathbb Z, \,a\in\{0,1,2,3,4,5,6\}$$ $a$ here is the remainder when $n$ is divided by $7$. If the difference between two numbers $a$ and $b$ is divisible by $7$, then that means that their remainder when divided by $7$ is the same. In mathematical notation, let's suppose we have two numbers whose difference is divisible by $7$ and inspect their remainders:

Let $p\in \mathbb Z$.

Let $q \in \mathbb Z : 7 | (p-q)$

Then: $$q = p + 7l \quad \text{ for some } l\in \mathbb Z$$ $$\quad p=7k + a\quad \Leftrightarrow \quad q=7(k+l) + a$$

$q$ then equals some multiple of $7$ plus the remainder $a$ which is the same remainder that $p$ has. Thus any two numbers in $\mathbb Z$ whose difference is divisible by $7$ have the same remainder as eachother when divided individually by $7$.

The converse is also true. Therefore two integers $a$ and $b$ are in the same equivalence class if and only if they have the same remainder when divided by $7$.

It is easy to see from here that there are a finite number of equivalence classes-- specifically, there are $7$:

$0$ is in the equivalence class containing $\cdots,-7,0,7,14,\cdots$

$1$ is in the equivalence class containing $\cdots,-6,1,8,15,\cdots$

$2$ is in the equivalence class containing $\cdots,-5,2,9,16,\cdots$

And so on.

By convention (and simplicity), we name these equivalence classes "$0$" through "$7$".

I'm not sure if you've studied this far yet, but the structure formed by quotienting $\mathbb Z$ by this equivalence relation is denoted: $$\mathbb Z / R = \mathbb{Z}_7$$

This is called a modular ring and consists of only $7$ elements.