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R is a relation defined on the integers by $(a,b) \in R$ is $a^2-b^2$ and is divisible by 3.

I set a or b to zero to get all the negative and positive values in the equivalence class. Although I want to say it is $(...,-9n,-6n,3n,0,3n,6n,9n,...)$ for some integer n. But I do not think this is correct because if n = 1, 6n does not belong to the relation. What am I doing wrong?

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If $a=3n,b=3m$ and if $a=3n\pm1, b=3m\pm1$ – lab bhattacharjee Nov 21 '13 at 4:27

We have your equivalence relation R as $a \sim b$ if $3|(a^2 - b^2)$.

To find the class of elements equivalent to $0$, we need to set one of the elements to $0$ (just one because of reflexivity, symmetry, and transitivity, as this is an equivalence relation): suppose $a \sim 0$, i.e. $3|(a^2 - 0^2) \Longrightarrow 3|a^2$. Then by Euclid's Lemma, $3|a$. So the equivalence class of $0$ is the set of all integers that we can divide by $3$, i.e. that are multiples of $3: \{\ldots, -6,-3,0,3,6, \ldots\}$.

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