Answer     9

It is sufficient to check divisors of $\varphi(19)=18$. $$\begin{array}{lcrl} 5^2 &\equiv& 6&\pmod{19} \\ 5^3 &\equiv& -8&\pmod{19} \\ 5^6 &\equiv& 7&\pmod{19} \\ 5^9 &\equiv& 1&\pmod{19} \end{array}$$

Therefore the answer is $\boxed{9}$.