Question

Use a direct proof to show that the sum of two even integers is even.

Short answer

The sum of two even integers is even.

Step-by-step solution

Introduction

The purpose is to show that the sum of two even integers is even by direct prove.

Proving that sum of two even integers is even.

Definition of even integer:

If there is an integer k, then the integer n is even such that\(n = 2k\).

Consider a and b be two even integers. Therefore, by definition of even integer, the values of a and b are given as:

\(a = 2k\)and\(b = 2l\)

Here, k and l are some integers.

The sum of two integers a and b is given as:

\(a + b = 2k + 2l\)

\( = 2(k + l)\)

Since, k and l are integers and sum of two integers is also a integer, therefore,\(k + l\)is also an integer.

Hence, \(a + b\)is an even integer.

Therefore, sum of two even integers is even.