Let p be a prime satisfying p 3 (mod 4).

Let p be a prime satisfying p = 3 (mod 4). (a) Let a be a quadratic residue modulo p. Prove that the number b = a^p + 1/4 (mod p) has the property that b^2 = a (mod p). This gives an easy way to take squareroot s modulo p for primes that are congruent to 3 modulo p. (b) Use (a) to compute the following squareroot s modulo p. Be sure to check your answers. (i) Solve b^2 = 116 (mod 587). (ii) Solve b^2 = 3217 (mod 8627). (iii) Solve b^2 = 9109 (mod 10663)