Question

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

Short answer

The sum of two odd integers is even integer.

Step-by-step solution

Introduction

Consider the following statements:

Statement 1: Odd numbers can be represented as\(2a + 1\), where\(a \in Z\)

Statement 2: Numbers that are divisible by 2 are even numbers.

Using statements and calculation to prove

Let \(x\)and\(y\) be two odd numbers.

Then by the hypothesis given above,

\(x = 2m + 1\) for some \(n \in Z\) and

\(y = 2n + 1\) for some\(m \in Z\)

Now, adding \(x\)and\(y\)

\(x + y = 2m + 1 + 2n + 1\)

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

Clearly, \(x + y\)is divisible by ‘2’.

As a result, by statement 2 \(x + y\)is even integer.

Hence, proved that the sum of two odd integers is even integer.