This is the second part of a question that I don't get. First part was for proving it is an equivalent relation, which I did, but I have a hard time finding out it's equivalence classes. How should I go about finding them?
A method, or thought process would be highly appreciated because I want to learn.
In principle there could be only up to $7$ equivalence classes, since the group of integers mod $7$ has only seven equivalence classes.
We then can ask ourselves, which numbers (mod $7$), when squared, yield which of the seven potential squares. Well, $$ 0^2 \equiv 0\\1^2\equiv 1\\ 2^2 \equiv 4 \\3^2 \equiv 2 $$ and for the other three values, $$ (7-k)^2 \equiv (-k)^2 \equiv k^2 $$ So the four equivalence classes are $$ \{0\}, \{1,6\}, \{2,5\}, \{3,4\} $$