Idempotents in $\mathbb Z_n$

As often happens when dealing with $\mathbf{Z}_n$, the Chinese remainder theorem is your friend. If the prime factorization of $n$ is $$ n=\prod_i p_i^{a_i}, $$ then by CRT we have an isomorphism of rings $$ \mathbf{Z}_n\cong\bigoplus_i \mathbf{Z}_{p_i^{a_i}}. $$ Observe that the isomorphism maps the residue class of an integer $m$ (modulo $n$) to a vector with all the components equal to the residue class of $m$ (this time modulo various prime powers): $$ \overline{m}\mapsto(\overline{m},\overline{m},\ldots,\overline{m}). $$ So the residue class of $m$ is an idempotent if and only if it is an idempotent modulo all the prime powers $p_i^{a_i}$.

Let us look at the case of a prime power modulus $p^t$. The congruence $x^2\equiv x\pmod{p^t}$ holds, iff $p^t$ divides $x^2-x=x(x-1)$. Here only one of the factors of, $x$ or $(x-1)$, can be divisible by $p$, so for the product to be divisible by $p^t$ the said factor then has to be divisible by $p^t$. Thus we can conclude that $x\equiv 0,1 \pmod{p^t}$ are the only idempotents modulo $p^t$. Therefore we require that $$ m\equiv 0,1\pmod{p_i^{a_i}} $$ for all $i$. By CRT these congruences are independent for different $i$, so the number of pairwise non-congruent idempotents is equal to $2^\ell$, where $\ell$ is the number of distinct prime factors $p_i$ of $n$.

Hint $ $ Idempotents in $\:\Bbb Z_{ n}\:$ correspond to factorizations of $\:n\:$ into two coprime factors. Namely, if $\:e^2 = e\in\Bbb Z_n\:$ then $\:n\:|\:e(e\!-\!1)\:$ thus $\:n = jk,\ j\:|\:e,\ k\:|\:e\!-\!1,\:$ so $\:(j,k)= 1\:$ by $\:(e,e\!-\!1) = 1.\:$ Conversely if $\:n = jk\:$ for $\:(j,k)= 1,\:$ then by CRT, $\:\Bbb Z_n\cong \Bbb Z_j\times \Bbb Z_k\:$ which has nontrivial idempotents $\:(0,1),\,(1,0).\:$ It is not that difficult to explicitly work out the details of the correspondence. Some integer factorization algorithms search for such nontrivial idempotents.

This correspondence between idempotents and factorizations holds more generally at the structural level - see the Peirce Decomposition. and this answer.

By the Chinese remainder theorem $\mathbf Z/n\mathbf Z$ is a product of more than one (unital) ring if and only if $n$ has more than one prime factor, and in this case $\mathbf Z/n\mathbf Z$ certainly has nontrivial idempotents. If on the other hand $n=p^k$ is a prime power, then all elements are either invertible (if not divisible by $p$) or nilpotent (if divisible by $p$) and this excludes the possibility of nontrivial idempotents. The case $n=1$ is a special case of a prime power (but the unique element now is both invertible and nilpotent; there are still no nontrivial idempotents of course).