(40 points) In each part, use the given method to prove/ disprove the statements: (a) (5 points) Disprove the following statement by nding an appropriate counterex- ample: For every real numbers :13, y, and 2, if a: > 0, y > O, and z > 0, then 10'3z+a:2+y2+z2 s (m+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 p 9.\" (c) (5 points) Prove that 017461461... is rational (digits 461 in the fractional part are periodically repeated forever). (d) (10 points) Use the method of direct proof to prove the following statement: Vm, n,l E 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