Find the equivalance classes of the relation defined by $a\,\rho\,b \, \text{iff} \, a^2-b^2 \,\text{is divisible by} \,7$

Question

Find the equivalance classes of the relation defined by $$a\,\rho\,b \, \text{iff} \, a^2-b^2 \,\text{is divisible by} \,7$$

I can show that it is a equivalance relation, but unable to find the classes. Please help.

Answer 1

Since $a^2\equiv _7 0,1,2,4$ we have 4 equivalence classes.

$C_1=\{a\in \mathbb{Z}; a \equiv_7 0 \}$

$C_2=\{a\in \mathbb{Z}; a \equiv_7 \pm 1 \}$

$C_3=\{a\in \mathbb{Z}; a \equiv_7 \pm 2 \}$

$C_4=\{a\in \mathbb{Z}; a \equiv_7 \pm 3 \}$