I been surfing all related stuff concerning solving indices but all I got are congruence solved using indices and I don't even know if that's the one i'm looking for. I'm trying to solve this question:
Can anyone point me out to the right direction or maybe give a sample demo? Thanks.
The index is like a logarithm. Typically the base (subscript) will be a primitive root of the modulus. The index of a number is also called the discrete logarithm.
$\text{ind}_a b \pmod{p}=c\Leftrightarrow a^c\equiv b \pmod{p}$.
So for instance $\text{ind}_{13}2 \pmod{19}=11$ because $13^{11}\equiv 2\pmod{19}$.
To do the other ones, you may want to make a table of powers of $13$ reduced $\pmod{19}$.