(1) Give a direct proof that the sum an even integer and an odd integer is odd. Hint: Start by letting m be an even integer and letting n be an odd...

(1) Give a direct proof that the sum an even integer and an odd integer is odd. Hint: Start by letting m be an even integer and letting n be an odd integer. That means m = 2k for some integer k, and n = 2j +1 for some integer j. You are interested in m + n, so add them up and see what you get. Why is the thing you get an odd integer?