RO-NMO 2021 Problem 123

Given is an positive integer $a>2$ a) Prove that there exists positive integer $n$ different from $1$ , which is not a prime, such that $a^n=1(mod n)$ b) Prove that if $p$ is the smallest positive integer, different from $1$ , such that $a^p=1(mod p)$ , then $p$ is a prime. c) There does not exist positive integer $n$ , different from $1$ , such that $2^n=1(mod n)$