Consider the following statements:

$S_1$: For any integer $n>1$, $a^{\phi(n)}\equiv 1\ (\text{mod }n)$ for all $a\in Z_n^*$, where $\phi(n)$ is Euler's phi function.

$S_2$: If $p$ is prime, then $a^p\equiv 1\ (\text{mod }p)$ for all $a\in Z_p^*$.

Which one of the following is/are correct?

Statement $S_1$ is true by Euler's theorem.

If $a\in Z_n^*$, then $a$ and $n$ are relatively prime.

So,

$a^{\phi(n)}\equiv 1\ (\text{mod }n)$

Now consider $S_2$.

For prime $p$, Fermat's theorem gives

$a^{p-1}\equiv 1\ (\text{mod }p)$

for $a\in Z_p^*$.

But $S_2$ says

$a^p\equiv 1\ (\text{mod }p)$

This is not generally true.

Actually,

$a^p\equiv a\ (\text{mod }p)$

So, $S_2$ is false.