Problem 1.1. The solution of x + 0 = 0 is z = 0, which is 0 + 0; the solution of 3: Problem 1.2. The solution of x2+ 1 = 0 is II: = 1, and this is a double root, The two solutions of x2 + J: = 0 are z = 0 and J: = 1. Since x 2 + x + 1 = 1 for both x = 0 and z = 1, the equation x2 + z + 1 = 0 has no solutions. Problem 1.3. If m = 37-+ 1 and n = 3s + 2, then m + n = 3(r + s + l), so that the remainder when m + n is divided by 3 is 0. Similarly, if m = 37-+ 2, then m + n = 3(r + s + 1) + 1, and so the remainder when rn + n is divided by 3 is 1.