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 bc 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, 1x + 31-x > 2