I need help with this question. Please thoroughly explain what you did for each step and labelame each proof that you used for each line. Essentially, just be as detailed as possible, if that's ok.

1. (40 marks) Prove the following statements. Explicitly state the type of proof you are using and clearly describe each step of the proof. (a) (5 marks) If x and y are both odd integers, then x+y is an even integer. (b) (5 marks) If x and y are positive, even integers, then x x y is divisible by 4 (c) (10 marks) (x mod 2)x (y mod 2) = (x x y) mod 2, where the mod operator gives you the remainder of the division (e.g. 3 mod 2 -1) (d) (5 marks) For any integers, x and y, X ye7 implies that x 2 4 or y 2 4 (e) (5 marks) For every integer n, n2 n is even. ( (10 marks) If x is a rational number and y is an irrational number, then the sum of x and y is irrational