Part II: Constructing proofs (60 pt.) You must write down all proofs in acceptable mathematical language: make sure you mark the beginning and end of the proof, define all variables, give a justification for every assertion (e.g., by definition of...), and use complete, grammatically correct sentences. Definitions: An integer n is even if and only if there exists an integer k such that n = 2k. An integer n is Two integers have the same parity when they are both even or when they are both odd. Two . An integer n is divisible by an integer d with d 0, denoted d | n, if and only if there exists an . A real number r is rational if and only if there exist integers a and b with b 0 such that odd if and only if there exists an integer k such that n 2k 1 integers have opposite parity when one is even and the other one is odd. integer k such that ndk r-a/b For any real number x, the absolute value of x, denoted x, is defined as follows: 2. (60 pt., 10 pt. each) Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof a. The product of any two even integers is divisible by 4 b. The product of any non-zero rational number and any irrational number is irrational. c. For all integers a, b, and c, if be is not divisible by a, then b is not divisible by a d. There is no integer that is both even and odd. e. Any two consecutive integers have opposite parity. f. For all real numbers x, x 31-x > 2