Mechanical problems step by step short but meaningful

. You may assume any of the following without proof: the laws of algebra are valid, the sum or difference of two integers is an integer, the product of two integers is an integer. . Additionally, you may use any facts about modular arithmetic/ algebra without jus- tification, such as the rules of modular addition and multiplication. This includes facts about algebra on even and odd numbers, such as even + odd = odd, even . odd = even, etc. (since these are equivalent to addition and multiplication mod 2). Definitions and Notation . An integer x is divisible by d if there exists an integer k with x = dk. . An integer x is rational if there exist integers a, b with x = a/b. Otherwise, it is irrational. . An integer x is prime if x 2 2, and there do not exist integers j 2 2, k 2 2 with x = jk. . An integer x is even if there is an integer k with x = 2k, or odd if there is an integer k with x - 2k + 1. If convenient, you may also use that x is even if x = 0 (mod 2), or odd if x = 1 (mod 2). Mechanical Problems 1. Even Stevens [14 points] Consider the proposition: for all integers x, y, if 3xy + 4x + y is odd then x is odd or y is odd. Give three separate proofs, using: (a) a proof by contrapositive (b) a proof by contradiction (c) a proof by cases on whether x is even or odd