Prove that if a and b are integers of the form 4n+1, then ab is of the form 4n+1.

Prove or disprove: If a and b are odd integers, then 4 | (a – b) or 4 | (a+b). -

Prove or disprove: For any integer a, 14 | (3a — 5) and 21 | (2a+3) .

Consider the following proof that if x and y are two even integers, then their sum is even. Proof: Assume that x and y are two even integers but their sum is not even, then x + y is odd. By definition, there are some integers p, q, and k such that x = 2p, y = 2q, and x + y = 2k + 1. Therefore, 2k + 1 = x + y = 2p + 2q. Rewrite above equation, we have 2(p + q - k) = 1, which is impossible. Which proof method is used in above proof. O Proof by counterexample Direct proof Proof by contraposition O Proof by contradiction