Even Numbers

Introduction

Even numbers are those that can be evenly divided into two equal groups or pairs and are divisible by `2`. Examples include `2`, `4`, `6`, `8`, and `10`. These numbers can be paired evenly, unlike numbers such as `5`, `7`, `9`, or `11`. Hence, `5`, `7`, `9`, or `11` are not considered even numbers. Let’s explore further about even numbers, their distinctions from odd numbers, along with some examples of even numbers. 

Definition of Even Numbers

An even number is a number that can be evenly divided by `2`. It means when you divide the number by `2`, there's no remainder. The last digit of an even number is `0`, `2`, `4`, `6`, or `8`. For instance, `6`, `10`, `14`, `18`, and so on are even numbers. Another way to think about even numbers is that they can be split into two equal parts. 

For example, `6` can be split into `3 + 3`.

Even and Odd Numbers

Odd numbers are those that can be represented as `(2n + 1)` where n is any natural number. They are not divisible by `2`. Examples of odd numbers include `27, 155, 63, 91`, and so on. On the other hand, even numbers can be expressed as `2n`, where `n` is a natural number, and they are divisible by `2`. Examples of even numbers are `12, 36, 48, 82, 100`, and so forth.

Difference Between Odd and Even Numbers

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

Properties of Even Numbers

There are three properties of even numbers listed below:

• Property of Addition
• Property of Subtraction
• Property of Multiplication

`1`. Property of Addition

`2`. Property of Subtraction

`3`. Property of Multiplication

What Are Even Prime Numbers?

The only prime number among even numbers is `2`. Unlike `2`, other even numbers have additional factors besides `1` and the number itself. For instance, the factors of `6` are `1, 2, 3`, and `6`. Therefore, `2` stands as the sole even prime number.

Consecutive Even Numbers

Consecutive even numbers are pairs of even numbers that follow each other in sequential order, with a difference of two. For instance, if we start with the even number `6`, the next consecutive even number would be `8` because it follows `6` and is two more. Similarly, `12` and `14` are consecutive even numbers because they are both even numbers and are two units apart. Another example is `18` and `20`, which also follow each other consecutively with a difference of two.

Solved Examples

Example `1`: Are the following numbers even or odd?

`i) 29`          `ii) 46`          `iii) 57`

Solution:

`i)` `29` is an odd number because its last digit, `9`, is not even. 

Additionally, it cannot be divided into equal parts.

Hence, `29` is an odd number.

`ii)` `46` is considered an even number because its last digit, `6`, is also an even number. 

Moreover, `46` can be divided into two equal parts.

Hence, `46` is an even number.

`iii)` `57` is an odd number because its last digit, `7`, is not even. 

Additionally, it cannot be divided into equal parts.

Hence, `57` is an odd number.

Example `2`: Sara bought `42` cookies for a party. She gave away `24` cookies to her friends. Will she have an even number of cookies left or an odd number of cookies left?

Solution: 

Sara gave away `24` cookies out of `42` cookies.

So, `42 - 24 = 18`

Now, Sara is left with `18` cookies.

We know that `18` is a multiple of `2`.

So, `18` is an even number.

Therefore, Sara will have an even number of cookies left, that is `18`.

Example `3`: Mark purchased `24` pencils for his two classmates. Can he distribute them equally among his classmates?

Solution:

Number of pencils `= 24`

`24` is considered an even number because its last digit, `4`, is also an even number. 

Moreover, `24` can be divided into two equal parts.

Hence, Mark can distribute `24` pencils equally between his two classmates.

Example `4`: Identify the even numbers from the given list.

`17, -22, 45, 68, -33, 50, 91, -14, 37, 82`

Solution:

Even numbers are numbers divisible by `2` and their last digit can be "`0`", "`2`", "`4`", "`6`", or "`8`".

Here the even numbers are `-22, 68, 50, -14` and `82`.

Example `5`: Find the sum of even numbers from `1` to `15`.

Solution:

Even Numbers from `1` to `15` are `2, 4, 6, 8, 10, 12, 14`.

Sum `= 2 + 4 + 6 + 8 + 10 + 12 + 14`

Sum `= 56`

Thus, the sum of even numbers from `1` to `15` is `56`.

Practice Problems

Q`1`. Calculate the sum of even numbers from `1` to `20`.

1. `100`
2. `120`
3. `110`
4. `130`

Answer: c

Q`2`. Which number is an even number?

1. `17`
2. `28`
3. `35`
4. `43`

Answer: b

Q`3`. Emma bought `75` cookies for a bake sale. She gave away `39` cookies to her friends. Will she have an even number of cookies left?

1. Yes
2. No
3. Cannot determine
4. None of the above

Answer: a

Q`4`. Select the even number.

1. `51`
2. `64`
3. `73`
4. `89`

Answer: b

Q`5`. Alex has `50` cookies to distribute among `5` guests. Can he distribute them equally?

1. Yes
2. No

Answer: a

Frequently Asked Questions

Q`1`. What defines an even number? 

Answer: Even numbers are integers divisible by `2`, leaving no remainder.

Q`2`. Are all numbers ending in "`0`" considered even? 

Answer: Yes, numbers ending in "`0`" are even, as they are divisible by `2`.

Q`3`. Can odd numbers be expressed as the sum of two equal parts? 

Answer: No, odd numbers cannot be evenly divided into two equal parts.

Q`4`. Are there any prime even numbers other than `2`? 

Answer: No, `2` is the only prime even number.

Q`5`. How are even numbers represented in pairs? 

Answer: Even numbers can be paired up neatly, as they are divisible into two equal groups.