Which of the following statements about Fermat's Little Theorem are TRUE?

(I) If $p$ is prime and $\gcd(a,p)=1$, then $a^{p-1} \equiv 1 \pmod{p}$

(II) For any prime $p$ and any integer $a$, we have $a^p \equiv a \pmod{p}$

(III) If $p$ is an odd prime and $\gcd(a,p)=1$, then $a^{(p-1)/2} \equiv \pm 1 \pmod{p}$