please answer questions #7-13

7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n + 2 is even, then n is even using proof by contradiction. 11. Prove that for every n E Z,n2 2 is not divisible by 4. Use proof by Contradiction. 12. Prove that if a and b are consecutive integers then the sum a + b is odd. Use proof by contrapositive. 13. Let m,n 2 0 be integers. Prove that if m+n 229 then m 2 15 or n 215. Prove by contradiction