Problem 2: Consider the set of integers. 1. Use a direct proof to show that the sum of two even integers is even. 2. Prove that if m+ n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind if proof did you use? 3. Use a direct proof to show that every odd integer is the difference of two squares. 4. Prove that if n is an integer and 3n + 2 is even, then n is even using a) A proof by contraposition. b) A proof by contradiction. 5. Prove that if n is an integer, the following four statements are equivalent: a) n is even. b) n+1 is odd. c) 3n + 1 is odd. d) 3n is even. 6. Find a counterexample to the statement Every positive integer can be written as the sum of the squares of three integers