Chapter 1.5, Problem 56E

To determine

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

Answer to Problem 56E

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

Explanation of Solution

Given information: Both $x$ and $y$ are odd.

An odd number is a number which is not divisible by 2. An odd number can be expressed in the form $2k+1$ , $k$ being an integer. Let $x=2m+1$ and $y=2n+1$ , $m$ and $n$ being integers.

Therefore,

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

Let, $m+n+1=p$

Hence, $x+y=2p$ . Since $2p$ denotes an even number, the sum of $x$ and $y$ is an even number.

For example:

$31+15=46$