Let \(p\) be a prime number and \(k\) a positive integer.

(a) Show that if \(p\) is odd and \(x\) is an integer such that \(x^{2} \equiv 1 \bmod p^{k}\), then \(x \equiv \pm 1 \bmod p^{k}\).

(b) Find the solutions of the congruence equation \(x^{2} \equiv 1 \bmod 2^{k}\). (Hint: There are different numbers of solutions according to whether \(k=1\), \(k=2\) or \(k>2\).)