Quotient set cardinal in $\mathbb{Z}_{12}$

Question

In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$

Then what is the cardinal of the quotient set?

Answer 1

Since there are only 12 elements, you can simply check this element by element. Clearly $\bar{0},\bar{1},\bar{2},\bar{3}$ form distinct equivalence classes since their squares are all different. Since $\bar{4}^2=\bar{2}^2, \bar{5}^2 = \bar{1}^2, \bar{6}^2=\bar{0}^2, \bar{7}^2=\bar{1}^2,\bar{8}^2=\bar{2}^2,\bar{9}^2=\bar{3}^2,\bar{10}^2=\bar{2}^2,\bar{11}^2=\bar{1}^2$, we know that this is a complete list. Therefore there are only 4 equivalence classes.