For any prime $p ≠ 2,5$, prove there are at most four values of the last digit of any power $p^{i}$?

Question

I am currently working on this question and I am thoroughly stuck. I believe that this question is saying that for any prime $p$, there will be four or less numerals $p-1$ that exist in the numeral $p$. Inductively we could show this is true for $n ≥ 2$ but I'm not sure if I am on the right track or not.

Answer 1

Hint If $gcd(p,10)=1$ then, as $\phi(10)=4$ we have $$p^{4} \equiv 1 \pmod{10}$$

Use this to show that the powers of $p$ $\pmod{10}$ repeat after 4 steps.

Answer 2

(Hint for someone who doesn't know what $\phi$ means)

When you rule out even numbers and multiples of $5$, what could the ones digit be?