RO-NMO 2008 Problem 123

Let $A$ be a unitary finite ring with $n$ elements, such that the equation $x^n=1$ has a unique solution in $A$ , $x=1$ . Prove that

a) $0$ is the only nilpotent element of $A$ ;

b) there exists an integer $k\geq 2$ , such that the equation $x^k=x$ has $n$ solutions in $A$ .