 ## Even Numbers

Even Numbers: A Whole number can be classified into even and odd numbers. If a number when divided by 2 leaves no remainder, then the number is called an even number. If the whole number when divided by 2 leaves any remainder then it is an idd number. To understand even numbers in a much easier and better way, you must know the tables/multiples of 2 and basic methods of division. For Example:

Let's talk about the numbers 4, 19, 54, and 488.

You can easily find out that 4 is an even number if you know the tables of 2. In the same way, you can easily find out that 19 is not an even number, as it is not divisible by 2. 19÷2= 9 leaving 1 as a remainder or quotient.

Now, if we have some numbers like 54 and 488,

54÷2= 27                  488÷2=244

Both the numbers are completely divisible by 2, leaving no remainder. So in the above examples, only 19 is not an even number. You can identify an even number more easily by going through this article. In this lesson, we will cover all about even numbers, definitions, examples, properties, and important points.

## Even Number Definition

An even number is defined as any number that is completely divided by 2, or the number that can be divided into two equal groups or pairs, without having any remainder, is called an even number. Now, Some examples are explained below. Check if the number is even or not.

8, if we divide 8 by 2 i.e 8 ÷ 2=4 and the remainder is 0. So the given number 8 is an even number.

12, if we divide 12 by 2 i.e. 12 ÷ 2 =6 then again the remainder is 0. So the given number 12 is an even number.

15, if we divide 15 by 2 i.e. 15 ÷ 2 then the remainder comes out as 1, so the given number 15 is not an even number. It is an odd number.

21, If We check the divisibility of 21 by the same method i.e 21 ÷ 2 gives the remainder 1. So 21 is not an even number. It is an odd number.

As you have noticed, those numbers which are not divisible by 2, leave a remainder hence they are an odd number instead of an even number.

An even number can also be defined as even an integer that is in the form n=2k, where k is an integer.

## How to find Even Numbers easily?

The easiest way to identify whether a given number is even or not. We must first look at the last digit of the given number:

1. If the last digit of a number ends up with 0, 2, 4, 6, or 8. It is even.
2. If the last digit of a number ends up with 1, 3, 5, 7, or 9. It is odd.

For example,

58 is an even number as the last digit of 58 is 8.

366 is an even number as the last digit of 366 is 6.

987654 is also an even number as the last digit of 987654 is 4.

23 is not an even number as the last digit of 23 is 3, so 23 is an odd number.

487 is also not an even number as the last digit is 7, and so it is also an odd number.

## Properties of Even Numbers

a) The addition of two even numbers will always result in an even number.

For example, 8+6=14, 14+82=96, 12+6=18

b) The addition of an even number and an odd number or vice versa results in an odd number.

For example, 4+9 =13, 11+10=21, 22+3=25

c) The sum of two odd numbers will always result in an even number.

For example, 7+3 =10, 9+9=18, 43+65=108

2. Properties of Subtraction:

a) The difference between two even numbers results in an even number.

For example, 16-4=12, 88-8=80, 442-322=120

b) The difference between an even number and an odd number or vice versa results in an odd number.

For example, 32-9=23, 19-10=9, 40-7=33

c) The difference between two odd numbers results in an even number.

For example, 7-3=4, 11-9=2, 47-5=42

3. Properties of Multiplication.

a) The multiplication of two even numbers is always an even number.

For example, 8 X 2=16, 14 X 4=56, 2 X 2=4

b) The multiplication of an even number and an odd is an even number.

For example, 4 X 7=28, 3 X 6=18, 12 X 5=60

c) The multiplication of two odd numbers is always an odd number. Sometimes this will not be considered as the properties of even numbers.

For example, 5 X 5=25, 11 X 11=121, 13 X 3=39

## Even Prime Numbers

Do you know a prime number has only two factors that are 1 and itself. Whereas an even number is divisible by 2. If a number is considered to be an Even Prime Number, It must fulfill both the above criteria. Hence, only 2 is the number that can be an even prime number. As all the other even numbers are divisible by 2, they cannot be prime numbers

Important points on Even Numbers:

1. In counting, every alternate number is an even number starting from 2 and an odd number starting from 1.
2. 0 ‘Zero’ is an even number.
3. The smallest even number is 2.
4. The only even prime number is 2.

## Even Numbers- Solved Questions

Q1: How to find out the sum of n even numbers?

Answer: The sum of n even numbers is calculated by the formula S=n (n+1)

Q2: Find out the sum of the first ten even numbers.

Answer: The sum of the first ten even numbers is calculated using the formula S=n(n+1)

So, here we have to find the sum of the first ten even numbers i.e n=10

By putting, n=10 in formula

S=10(10+1)

=10(11)

Now, we can also cross-check the solution as

The first 10 even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20.

2+4+6+8+10+12+14+16+18+20=110 hence the sum of the first ten even numbers is 110

Q3: Find out the even number from the given set of numbers 633, 524, 873, 448, 87, 222, 65, 14872, 8945, 25648.

Answer: The even numbers from the list are: 524, 448, 222, 14872, and 25648.

So, the total number of water bottles in their house is 84 (even) + 60 (even) = Even i.e 144

Q4. Write any eight consecutive even numbers between 50 to 70.

Answer: All the numbers between 50 to 70 are 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,62, 63, 64, 65, 66, 67, 68, 69, and 70. The eight consecutive even  numbers between 50 to 70 are 52, 54, 56, 58, 60, 62, 64, and 68.

Q5: Kamal buys 84 new water bottles and his brother Satyam also brings 60 new water bottles to his home. Is the total number of water bottles in their house an even number?

Answer: The number of water bottles bought by Kamal is 84, an even number.

The number of water bottles brought by Satyam is 60, which is also an even number.

So, the total number of water bottles in their house is

84 (even) + 60 (even) = Even i.e 144

Q6: Find the sum of all the even numbers from 1 to 200?

Answer:  There are a total of 100 even numbers between the numbers 1 to 200.

Here, n = 100

By using the formula S= n(n+1), we can calculate the sum of all the even numbers present between 1 to 200.

Since, n=100

S = n(n+1)

S = 100(100+1)

S = 100(101)

So, here S= 100 X 101

=10100

Hence, the sum of the even numbers from 1 to 200 is 10,100.

Q7: Which is a correct statement?

1.  The Product of two even numbers is always an even number.
2.  The Sum of two even numbers is always an odd number.

Here, if we consider statement 1, i.e Even X Even = Even which is right. As for example, 6 X 6 =36, 8 X 6 = 48.

But, statement 2 is incorrect as Even + Even = Even,

For example, 4+8=12, 10+12=22. After adding two even numbers the sum is also an even number, not odd.

Q8: Choose the correct option. The difference between two even numbers is:

a) Always an even number

b) Always an odd number

c) Can be both odd and even

d) None of the above is correct.

Answer: The correct answer is option a). As Even number - Even number = Even. For example, 8 - 4= 4, 18 - 6 =12

Answer: Yes, Zero (0) is an even number. An even number must always be divisible by 2. So, When 0 is divided by 2, the resulting quotient remainder is also Zero which is an integer, thereby classifying it as an even number.

Q10: What is the sum of 4 consecutive even numbers between 20 to 30.

Answer: The Four consecutive even numbers between 20 to 30 are:

22, 24, 26, and 28. So, the sum of the first four consecutive even numbers between 20 to 30 is 22 + 24 + 26 + 28 = 100.

Even Numbers: FAQs

Ans: An even number is defined as any number that is completely divided by 2.

Ans: 2 is the only even prime number. As all the other even numbers are divisible by 2, they cannot be prime numbers.

Ans: If the last digit of a number ends up with 0, 2, 4, 6, or 8. It is an even number.

Ans: The general form of an even number is n=2k, where k is an integer.

Ans: 100- even numbers are in between 1 to 200.

Ans: Yes, zero is an even number. Usually, zero together with the even numbers, so certainly it is not an odd number.

Ans: There are 50 even numbers between 1 to 100.