Prove each of the following statements. (a) The sum of two even integers is always even. (b) The sum of an even integer and an odd integer is always odd.

Question

Quizplus · Accepted Answer

The Answer is (a) Suppose that m and n are even integers. Then there are integers j and k such that m=2 j and n=2 k. It follows that m+n=2 j+2 k=2(j+k)=2 l, where l=j+k. Hence m+n is even.
(b) Suppose that m is even and n is odd. Then there are integers j and k such that such that m=2 j and n=2 k+1. It follows that m+n=2 j+(2 k+1)=2(j+k)+1=2 l+1, where l=j+k. Hence m+n is odd.

Prove Each of the Following Statements