How to prove that the number of unique prime factors is the number of idempotent elements contained in $\mathbb Z_n$

Question

Since the number of unique prime factors is connected to the number of idempotent elements contained $ \mathbb Z_n$, Denoted as; $ | I_n |= 2^j$ , where j is number of unique prime factors contained in $\mathbb Z_n$ Example: $ \mathbb Z_ 6 = \{0,1,2,3,4,5\}$ the idempotent elements are $I_6 = \{0,1,3,4\}$ By unique prime factorization $n=(p_1)^k(p_2)^k...(p_j)^k$,then 6= (2)(3). Therefore the generalization on getting the number of idempotent elements in $\mathbb Z_n$ is ; $ | I_n |= 2^j$ how do I formally prove this