(40 points) In each part, use the given method to prove/disprove the statements: (a) (5 points) Disprove the following statement by finding an appropriate counterexample: For every real numbers x, y, and z, if x > 0, y > 0, and z > 0, then 103z + x2 + y2 + z2 (x + y + z)2. (b) (5 points) Use the method of exhaustion to prove the following statement: "For every prime number p between 30 and 58, 10 does not divide p9." (c) (5 points) Prove that 0.17461461 . . . is rational (digits 461 in the fractional part are periodically repeated forever). 2 (d) (10 points) Use the method of direct proof to prove the following statement: m, n, l Z,(m is even n is odd) (ml + n is odd). (e) (15 points) Use the method of direct proof to show that for any positive 5-digit integer n, if n is divisible by 9, then some of its digits is divisible by 9 too