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

Question

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 $\text{ind}_3(20)$

Answer 1

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.