4. Use a proof by contrapositive to prove that, for integers r and y: if a+y is odd, then exactly one of r and y is odd and the other is even.

Use the method of proof by contradiction to prove that their exist no integers a and b for when 18a + 6b = 1

1. How many solutions (ordered 4-tuples (x1, x2, ¤3, x4)) are there to the equation Xị + x2 + 13 + x4 = 16 where x;, i = 1, 2, 3, 4 is a nonnegative integer such that x1 > 1, x2 > 2, x3 > 3 and x4 2> 4 ? Use Mathematica to generate them all.

5. Prove each of the following statements:.. (a) If m, n are odd integers, then mn is an odd integer. (b) If m, n are integers and mn is even, then m is even or n is even. (c) If z, y are real numbers and x +y is irrational, then z is irrational or y is irrational.