Give the equivalence classes of the relation aRb if and only if $a^4 ≡ b^4 \pmod {30}$ on the set $\{1,2,3,...,15 \}$

Question

Give the equivalence classes of the relation $aRb$ if and only if $a^4 ≡ b^4 \pmod {30}$, on the set $\{1,2,3,\dots ,15 \}$.

I have looked online to understand what this question means but I really do not understand how I should proceed to solve it. Thank you for any help.

Answer 1

The equivalence class of a $x \in A$ ($A$ is a set) is defined as: $$[x] = \{a \in A | aRx \}$$ Meaning it is the set of all items in $A$ related to $x$. Your question is to find all of those equivalence classes to the given relation.

So starting with the equivalence class of $1$: We need to find all $ 1 \le x \le 15$ such that $x^4 ≡ 1 \pmod {30}$. An example is $x = 7$. Can you continue?