Primitive roots and Discrete logarithms question

Question

Can anyone help me solve this previous exam question?

Answer 1

Let $g$ be a primitive root mod 101. Then as $g^0,g^1,...,g^{99}$ runs through all nonzero residues modulo 101, $a$ has a solution in $(\star)$ iff there is $n$ with $a=(g^{n})^{15} = g^{15n}$. But by Fermat's Little Theorem, $g^{15n}$ has $\frac{100}{gcd(100,15)}=20$ different values mod 101. This gives the answer for a).

For b) you may show that if $a$ has a solution $x$, then $y$ is a solution to $(\star)$ iff $y=xg^{20n}$ for $n\in \Bbb Z$.

For the nontrivial direction of c) you can show similiarly to a) that there are exactly 20 solutions to $a^{20}=1$. Because of a), these must all have solutions to $(\star)$.

But in general, things get a bit easier if you start by showing that the solutions to $x^{15}=a$ have a 1:1-correspondence to those of $x^5 = b$ with $b^3=a$ by using that $x \rightarrow x^3$ is 1:1 in $(\Bbb Z / 101 \Bbb Z)^x$.