Chapter 1.5, Problem 55E

To determine

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

Answer to Problem 55E

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

Explanation of Solution

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

A number which is divisible by $2$ is an even number.Since both $x$ and $y$ are even numbers, $2$ will be the common factor for both the numbers and hence $2$ will also be a factor for the sum. An even number can be expressed in the form $2k$ , $k$ being any integer. Let, $x=2m$ and $y=2n$ , $m$ and $n$ being factors of $x$ and $y$ respectively.

$\begin{array}{l}x+y=2m+2n\\ x+y=2(m+n)\\ \end{array}$ Let, $m+n=p$ .

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

For example:

  $24+12=36$