# Even Numbers: Definition, Properties, Examples

Numbers play a very crucial role in the world of mathematics, even in our daily life to count things. Note that 4 can be divided into two equal parts, but 5 cannot. Here 4 is an even number, but 5 is not. So we can say that numbers have types. There are two types of numbers: even numbers and odd numbers. In this section, we will learn about even numbers, their important properties, and key facts about them.

#### Definition of Even Numbers

A number is called an even number if it is completely divisible by 2. So even numbers leave the remainder 0 when we divide by 2. Note that 2 is the first positive even number and 0 is the first non-negative even number.

By the above definition, we can say that any even number is a multiple of 2. Moreover, all multiples of 2 are even numbers.

Examples of even numbers: 10, 16, 22, 32, 48, 100, 1000 are few examples of even numbers.

#### General Form of Even Numbers

As even numbers are completely divisible by 2, we can express an even number as 2k for some integer k. So the set of all even numbers is given below:

$\{2k : k \text{ is an integer}\}$

As $\mathbb{Z}:=$ $\{k: k \text{ is an integer}\},$ we obtain that $2\mathbb{Z}$ denotes the set of all even integers.

#### Is zero an even number?

Note that we can write 0=2×0. So the number 0 can be expressed as 2k where k=0. So by the definition of even numbers, we conclude that 0 is an even number.

#### Is one an even number?

Let us write 1=2k. This implies that k=1/2. So k is not an integer. Thus 1 is not divisible by 2. Hence 1 is not an even number.

#### List of Even Numbers from 1 to 100

The even numbers from 1 to 100 are given below:

2, 4, 6, 8, 10,12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98 and 100.

So there are 50 even numbers from 1 to 100.

#### How to check even numbers?

As even numbers are multiples of 2, so an even number must have a unit digit (last digit) either 0, 2, 4, 6 or 8. In this way, we can check a given number is even or not, no matter how much big the number is.

For example, the number 9876452 is an even number as it has the last digit 2. But the number 76987 is not an even number since the unit digit is 7.

#### Properties of Even Numbers

• Every even number ends with either 0, 2, 4, 6 or 8.

• Every even number is a multiple of 2.

• Even number + Even number = Even number

• Even number – Even number = Even number

• Even number × Even number = Even number

#### Addition or Subtraction of Even Numbers

The sum of two even numbers is always an even number.

Proof: Let a and b be two even numbers. So we have

a=2k  and b=2m  for some integers k and m.

a+b = 2k+2m = 2(k+m)

So a+b is a multiple of 2.

This implies that a+b is an even number, proving the statement.

In a similar way, we have a-b=2(k-m). So a-b is a multiple of 2. This makes that a-b is an even number. So we conclude the following:

The difference between two even numbers is again an even number.

#### Multiplication of Even Numbers

The product of two even numbers is an even number.

Proof: Let a and b be two even numbers. So we have

a=2k  and b=2m  for some integers k and m.

Multiplying a and b we get that

a × b = 2k × 2m = 2×2km

So ab is a multiple of 2.

This implies that ab is an even number, proving the statement

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