Chapter 1.5, Problem 57E

To determine

Whether the sum of $x$ and $y$ (integers) is even or odd.

Answer to Problem 57E

The sum of $x$ and $y$ is odd.

Explanation of Solution

Given information: $x$ is even and $y$ is odd.

An even number is expressed as $2k$ while an odd number is expressed as $2k+1$ , $k$ being any integer. Let, $x=2m$ and $y=2n+1$ , $m$ and $n$ being integers.

$\begin{array}{l}x+y=2m+2n+1\\ x+y=2(m+n)+1\\ \end{array}$

Let, $m+n=p$

Therefore, $x+y=2p+1$ . Since the term $2p+1$ denotes an odd number the sum will be odd.

For example:

$16+11=27$