Yesterday, my friend got a task from his lecturer. The task is to show whether there is a relationship between those two theorem or not. I thought there must be a relationship between these two, because it takes the same requirement ($a$ and $n$ [or $p$] must be coprime). But I don't know how to figure it out formally.
We know that $$a^{\phi(n)}\equiv 1\mod n$$ given $a$ and $n$ coprime. Then, if we take $n=p^k$ with $p$ prime and $k \in \mathbb{N}$, from Euler Phi Function of Prime Power we got
$$\phi(p^k)=p^{(k-1)}(p-1).$$
Finally, we got
$$a^{p^{(k-1)}(p-1)}\equiv 1\mod p^k \quad(1)$$
If we use $k=1$ on $(1)$, then
$$a^{(p-1)}\equiv1\mod p \quad(2).$$ Since $a$ is coprime with $p^k$, then $a$ is coprime with $p$, thus we got Little Fermat theorem.
My question is if $k>1$, is there any way to simplify $(1)$ to $(2)$? What theorem should be used?