Chapter 5.4, Problem 25PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $x$ is an odd number.

Prove: $x$ is not divisible by $4$ .

Calculation:

In an indirect proof or proof by contradiction, assume that the conclusion is false. By showing the assumption to be logically impossible, prove the assumption is false and the original conclusion is true. For this problem, assume that $x$ is divisible by $4$ .

Given: $x$ is an odd number.

Prove: $x$ is not divisible by $4$ .

Following steps are used for the indirect proof are as shown below:

Firstly, assume $x$ is divisible by $4$ . In other words, $4$ is a factor of $x$ .

Secondly, let $x=4n$ , for some integer $n\cdot x=2(2n)$

So, $2$ is a factor of $x$ which means $x$ is an even number, but this contradicts the given information.

Since the assumption that $x$ is divisible by $4$ leads to a contradiction of the given, the original conclusion that $x$ is not divisible by $4$ must be true.