listing elements of equivalence class

Question

For $m,n$ in $N$ define $m$ equivalent $n$ if $m^2-n^2$ is multiple of $3$ a) show that this is an equivalence relation b) list elements in equivalence class [0] c) list elements in equivalence class [1] d)do you think there are any more equivalence classes?

for part a i proved that its true by showing through matrix that its symmetric,reflexive and transitive. my problem is that i don't really understand how to find equivalence class, can someone explain how to do it and show on part b or c, rest ill do myself once i get how to do it.

Answer 1

The equivalence class $[0]$ is the set of all elements related to $0$. We have $m \in \mathbb{N}$ is related to $0$ $\iff m^2 - 0 = m^2$ is a multiple of $3$. Since $3$ is prime, $m \sim 0 \iff m$ is a multiple of $3$. Hence, $[0] = \{3k : k \in \mathbb{N} \}$.

Similarly, $m \sim 1 \iff m^2 - 1 = (m + 1)(m-1)$ is a multiple of $3$, in which case $m \equiv 1, 2 \bmod 3$. So $[1] = \mathbb{N} \setminus 3\mathbb{N}$.

Every positive integer is congruent to $0, 1$, or $2$ modulo $3$, so there are no more equivalence classes.

Answer 2

There are exactly two equivalence classes:

a. The set of integers divisible by 3, and

b. The set of integers non-divisible by 3.

Clearly, if $3\mid m$ and $3\mid n$, then $3\mid m^2$ and $3\mid n^2$, and hence $3\mid m^2-n^2$.

If $3\not\mid m$ and $3\not\mid n$, then $m^2$ and $n^2$ leave remainder 1, when divided by 3, and hence $3\mid m^2-n^2$.