Modular inverses

I would think the answer to the last part is going to depend on your exact definition of $\mathbb{Z}_p^*$.

I suspect his definition would be (and I gave my answer):

$\mathbb{Z}_p^*=\{\bar{1}, \bar{2},...,\overline{p-1}\}$ where $\bar{i}$ is the equivalence class of i mod p. (I.e. the set of all integers congruent to i mod p).

But if op would clarify that would help a lot.

I suspect his definition would be (and I gave my answer):

$\mathbb{Z}_p^*=\{\bar{1}, \bar{2},...,\overline{p-1}\}$ where $\bar{i}$ is the equivalence class of i mod p. (I.e. the set of all integers congruent to i mod p or equivalently the set of integers that leave the same remainder as i when divided by p).

But if op would clarify that would help a lot.

Firstly it is given in the question that a is not equal to p since a is given to be in Z_p\{0} but p is not in Z_p\{0} (since p is congruent to 0 mod p and 0 is not in the set).

The size of Z_p\{0} can be found by considering how many remainders you can get when dividing by p that is you can get {1,2,3,...,p-1}. So it has size p-1.

In fact the set is isomorphic to the quotient group Z/pZ. Integers mod p.