Homomorphisms and Isomorphisms · Challenge problem

The multiplicative group $U(17)$ of nonzero residues mod $17$ is cyclic of order $16$ (isomorphic to $\mathbb Z_{16}$). For how many $k \in \{1, 2, \ldots, 16\}$ is the map $x \mapsto x^k$ an automorphism of $U(17)$?