Question

Show that the square of an even number is an even number using a direct proof.

Short answer

Square of an even number is also an even number

Step-by-step solution

Introduction

Consider an even number m.

The purpose is to show that square of m is also an even number.

Step 2

As m is an even number. Therefore, m can be written as,

\(m = 2r\), where r is an integer.

Squaring both sides and analyzing for the result

Squaring on both sides of the above equation. This gives,

\({m^2} = {\left( {2r} \right)^2}\)

\( = 4{r^2}\)

As square of an integer is also an integer. This gives,

\({m^2} = 2{\left( {2r} \right)^2}\)

This implies,\({m^2}\)is an even number.

Hence, square of an even number is also an even number.