Exercise 2.5.5: Proving statements using a direct proof or by contrapositive. About Prove each statement using a direct proof or proof by contrapositive. One method may be much easier than the other. (a) The product of any integer and an even integer is even. Solution (b) If p > 2 and p is a prime number, then p is odd. Solution (c) For every non-zero real number , if is irrational, then is also irrational. (d) If is a real number such that , then . Solution (e) If n and m are integers such that n2 m2 is odd, then m is odd or n is odd. (f) If x, y, and z are integers and x z and y z are both even, then x y is also even. (g) The difference of two rational numbers is a rational number. (h) If , then . (i) If x is an odd integer then is even.