Answer     16

It is sufficient to check divisors of $\varphi(17)=16$. $$\begin{array}{lcrl} 3^2 &\equiv& 9&\pmod{17} \\ 3^4 &\equiv& -4&\pmod{17} \\ 3^8 &\equiv& -1&\pmod{17} \\ 3^{16} &\equiv& 1&\pmod{17} \end{array}$$

Therefore the answer is $\boxed{16}$.