Chapter 5.4, Problem 24PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $n^2$ is even.

Prove: $n^2$ is 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 $n$ or $y$ is not divisible by $4$ .

Given: $n^2$ is even.

Prove: $n^2$ is divisible by $4$ .

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

Firstly, assume $n^2$ is not divisible by $4$ . In other words, $4$ is not a factor of $n^2$ .

Secondly, if the square of a number is even, then the number is also even. So, if $n^2$ is even, $n$ must be even.

Let $n=2a$

  $n=2a$

  $n^2={(2a)}^2\text{ or 4}{a}^2$

  $4$ is a factor of $n^2$ ,which contradicts the assumption.

Since the assumption that $n^2$ is not divisible by $4$ leads to a contradiction of the assumption, the original conclusion that $n^2$ is divisible by $4$ must be true.