Write the index table for the primitive root $3$ of $25$ 
Write the index table for the primitive root $3$ of $25$

My attempt:
$$
\begin{array}{c|lcr}
k & 0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\
\hline
3^k & 1&3&9&2&6&18&4&12&11&8&24&22&16&23&19&7&21&13&14&17 \\
\end{array}
$$
so the index table will be 
$$
\begin{array}{c|lcr}
a & 1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&23&24\\
\hline
\text{ind}_3 a & 0&3&1&6&\color{red}?&4&15&9&2&\color{red}?&8&7&17&18&\color{red}?&12&19&5&14&\color{red}?&16&11&13&10 \\
\end{array}
$$
As you can see I can't find $\text{ind}_3(5),\text{ind}_3(10),\text{ind}_3(15)$ and $\text{ind}_3(20)$

By saying $3$ is a primitive root of $25$ it only means that the positive integers less than 25 and relatively prime to it are all of the form $3^k$ for an appropriate exponent $k$ so your table is essentially right, only just exclude the $5,10,15,20$ because they are not coprime with 25.
