 # Even and Odd Numbers

Even Number: Any number which is exactly divisible by 2 is called an even number. i.e. if a number when divided by 2 leaves no remainder, then the number is called an even number.

Examples of Even numbers:  2, 4, 6, 8, 10, 12, 14, 42, 100, 398, 996 etc.

While,

Odd Number: The number which is not exactly divisible by 2 is called an odd number. i.e. they leave the remainder as 1 when divided by 2.

Examples of Odd numbers: 1, 3, 5, 7, 9, 11, 13, 25, 47, 99, 273, 999 etc.

Note: 0 (zero) is an Even number because when 0 divided by 2, the quotient is 0 and the remainder left after division is 0.

### How to Check if a Given Number is Even or Odd Without Dividing it by 2.

A number can be checked if it is even or odd without dividing it by 2. This can be done by just checking the units place of the given number.

If the units place of the given number contains digits 0, 2, 4, 6 or 8. Then the given number is an even number. While,

If the units place of the given number contains digits 1, 3, 5, 7 or 9. Then the given number is an odd number.

• Unit digit of Even number contains either of the following digits:  0, 2, 4, 6 or 8.

• Unit digit of Odd number contains either of the following digits: 1, 3, 5, 7 or 9.

By this method, we can check if the given large number is Even or Odd.

Let us consider two numbers- (1) 7580 and  (2) 2311.

It is clear that in the first number 7580 the unit digit is 0. So, 7580 is an even number.

And for the second number 2311, the unit digit is 1. So, 2311 is an odd number.

### Set Representation of Even and Odd Numbers

The sets of even and odd numbers are represented as follows:

• Even = {2k: k ϵ Z}

• Odd = {(2k + 1): k ϵ Z}; Where, ‘k’ belongs to an integer (Z).

A formal definition of an even number is an integer of the form n = 2k, where k is an integer. And an odd number is defined as an integer of the form n = 2k + 1, where k is an integer.

List of Even Numbers Up to 100.

There are 50 even numbers between 1 to 100. They are given in the below table.

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

List of Odd Numbers Up to 100.

There are 50 odd numbers between 1 to 100. They are given in the below table.

 1 11 21 31 41 51 61 71 81 91 3 13 23 33 43 53 63 73 83 93 5 15 25 35 45 55 65 75 85 95 7 17 27 37 47 57 67 77 87 97 9 19 29 39 49 59 69 79 89 99

Properties of Even and Odd Numbers

1. The sum or difference of two even numbers is an even number.

Example: 24 + 10 = 34; 18 – 6 = 12.

2. The sum or difference of two odd numbers is an even number.

Example: 11 + 9 = 20; 13 – 9 = 4.

3. The sum or difference of an even number and an odd number is an odd number.

Example: 25 + 18 = 43; 36 – 15 = 21.

4. The product of two even numbers is an even number.

Example: 14 x 2 = 28

5. The product of two odd numbers is an odd number.

Example: 7 x 7 = 49

6. The product of an even and an odd number is an even number

Example: 7 x 8 = 56; 16 x 3 = 48

7.An even number is divisible by 2, and leaves the remainder 0.

8. An odd number is not completely divisible by 2, and leaves the remainder 1.

Can a Decimal or Fraction be Even or Odd ?

The answer to this question is NO

Only integers can be classified as even or odd. Decimals and fractions can not be classified in the category of even or odd.

Therefore, we can not say that any decimal like, 5.47 is odd by just checking its last digit i.e, 7 as odd. and 23.04 is even as its last digit 4 is even.

Solved Examples:

Q.1.  Choose the Even and Odd Numbers From the Given Set of Numbers.

23, 09, 10, 78, 133, 386, 689, 1121, 1322, 2020.

Ans.  The even numbers in the given set of numbers can be identified by checking their units digit which should be either 0, 2, 4, 6 or 8.

So, even numbers are: 10, 78, 386, 1322 and 2020.

In the same manner odd numbers in the given set of numbers can be identified by checking their units digit which should be either 1, 3, 5, 7 or 9.

So, odd numbers are: 23, 09, 133, 689 and 1121.

Q.2. What is the Smallest Positive Even Number and an Odd Number?

Ans. The smallest positive even number is 2. And

The smallest positive odd number is 1.

Q.3. What will be the Resulting Number (Even or Odd) for the Following Operations?

a) 9875 + 3001

b) 6901 – 988

c) 5600 x 2021

d) 777 – 111

Ans. Using the properties of even and odd numbers, we obtain:

1. Even b) Odd c) Even d) Even