Discrete Math Prove or disprove the following statements The

(Discrete Math)

Prove or disprove the following statements. There exists a real number x such that x^2 > x + 9. For all integer x 1 there exists an integer y such that xy = 6x + y. Let m be an odd integer and n be an even integer. Prove that their sum is odd.

Solution

4)

5.

Suppose that m and n are arbitrary odd integers. Then m = 2a + 1 and n= 2b; where a and b are integers. Then

m+n = (2a + 1) + (2b) (substitution)

     = 2a+ 2b+ 1

     = 2(a+b) + 1 (distributive law)

Since m+n is twice an integer namely, 2(a+b) plus 1, m+n is odd