(a) Suppose that K = (a, b) is a key in an Affine Cipher over Z_n. Prove that K is an involutory key if and only if a-i mod n = a and b(a + 1)=0 (mod n).

(b) Determine all the involutory keys in the Affine Cipher over Z15.

(c) Suppose that n=pq, where p and q are distinct odd primes. Prove that the number of involutory keys in the Affine Cipher over Z_n is n +p+q+1.