Let p be an odd prime, let a be an integer that is not divisible by p, and let b be a square root of a modulo p. This exercise investigates the square root of a modulo powers of p. (a) Prove that for some choice of k, the number b + kp is a square root of a modulo p^2, i.e., (b + kp)^2 ≡ a (mod p2). (b) The number b = 537 is a square root of a = 476 modulo the prime p = 1291. Use the idea in (a) to compute a square root of 476 modulo p^2.

Let p be an odd prime, let a be an integer that is not divisible by p, and let b be a square root of a modulo p. This exercise investigates the square root of a modulo powers of p.
(a) Prove that for some choice of k, the number b + kp is a square root of a modulo p^2, i.e., (b + kp)^2 ≡ a (mod p2).
(b) The number b = 537 is a square root of a = 476 modulo the prime p = 1291. Use the idea in (a) to compute a square root of 476 modulo p^2.