I've noticed $|\mathbb{Z}_n^{\times}|$ is always even for $n\geq3$.
I've also observed that $|\mathbb{Z}_n^\times|$ is always even no matter whether $n$ is prime or not. When $n$ is prime and greater than 2, $|\mathbb{Z}_n^\times| = n-1$, which is even. If $n$ is not prime, then we have $\mathbb{Z}_n^\times = \{a \in \mathbb{Z}_n^\times | \gcd(a,n)=1\}$, and $|\mathbb{Z}_n^\times| = k|a|$. How can I tell that $2|k$?