**Introduction**

Mathematics is a world of numbers. We have different types of numbers such as natural numbers, whole numbers, decimal numbers, and fractions that are based on certain characteristics of numbers. All these types of numbers have one thing in common. They all have odd numbers and even numbers. So, what are odd numbers and even numbers and how do we categorise them? Let us find out.

**What are odd and even numbers?**

Consider the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 and 10. They are all-natural numbers. If you look at them closely, some of the numbers such as 2 , 4 , 6 , 8 and 10 are divisible by 2 while the remaining numbers 1 , 3 , 5 , 7 and 9 are not divisible by 2. So, can we form a group of numbers based on their divisibility by 2? Yes, this is how we define odd numbers and even numbers.

Even numbers are numbers that are divisible by 2, leaving a remainder 0. For instance, in the example above, 2 , 4 , 6 , 8 and 10 are even numbers.

Odd numbers are numbers that are not divisible by 2, and always leave a remainder 1 when divided by 0. For instance, in the example above, 1 , 3 , 5 , 7 and 9 are odd numbers.

**Odd Numbers and Even Numbers from 1 to 100**

Given below is an even and odd numbers chart. It shows numbers till 100. The numbers shaded in blue are even numbers whereas the numbers shaded in yellow are odd numbers.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |

71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |

81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

Do we always have look at the number table to identify whether the number is an odd number or an even number, or we can identify the number using any other method? Let us find out.

**How to Identify Even Numbers and Odd Numbers **

We now know that a number that is divisible by 2 and generates a remainder of 0 is called an even number. On the other hand, an odd number is a number that is not divisible by 2. Also, the remainder in the case of an odd number is always “1”. So, can we identify whether the number is an even number or an odd number just by looking at the number itself?

Let us consider the numbers from 1 to 10. We have 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Out of these we know that the numbers 1, 3, 5, 7 and 9 are odd numbers and the numbers 2, 4, 6, 8, 10 are even numbers. Similarly, let us now consider the numbers from 11 to 20. We have 11, 12, 13, 14, 15, 16, 17, 18, 18 and 20. Out of these 11, 13, 15, 17 and 19 are odd numbers while the numbers 12, 14, 16, 18 and 20 are even numbers? Do you see a pattern here? Let us write down the odd numbers and the even numbers separately. We will have

Odd Numbers – 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Even Numbers – 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Look at the last digits of the odd numbers. The odd numbers end up in any of the five digits 1, 3, 5, 7 and 9. Therefore we can say that all odd numbers have any of the five digits 1, 3, 5, 7 and 9 at their units place. **Hence, we can say that the numbers that have any of the five digits 1, 3, 5, 7 and 9 are all odd numbers. **

Now, look at the last digits of the even numbers. The even numbers end up in any of the five digits 0, 2, 4, 6 and 8. Therefore we can say that all even numbers have any of the five digits 0, 2, 4, 6 and 8 at their units place. **Hence, we can say that the numbers that have any of the five digits 0, 2, 4, 6 and 8 are all even numbers. **

Let us now look at the properties of odd numbers and even numbers.

**Properties of Odd Numbers**

- When we add two odd numbers, the result is always an even number. For example, when we add 7 and 15 we will get 7 + 15 = 22. Similarly, when we add 13 and 17, we will get 13 + 17 = 30 which is an even number.
- When we subtract two odd numbers, the result is always an odd number. For example, when we subtract 9 from 17 we will get 17 – 9 = 8. Similarly, when we subtract 21 from 37, we will get 37 – 21 = 16 which is an even number.
- When we multiply two odd numbers, the result is always an odd number. For example, if we multiply 3 which is an odd number by 5 which is again an odd number, the result will be 3 x 5 = 15 which is again an odd number. Similarly, if we multiply 7 which is an odd number by 9 which is again an odd number, the result will be 7 x 9 = 63 which is again an odd number.
- When we multiply an even number and an odd number, the result is always an even number. For example, if we multiply 3 which is an odd number by 6 which is an even number, the result will be 3 x 6 = 18 which is an even number. Similarly, if we multiply 9 which is an odd number by 4 which is an even number, the result will be 9 x 4 = 36 which is an even number.

**Properties of Even Numbers**

- When we add two even numbers, the result is always an even number. For example, when we add 8 and 6 we will get 8 + 6 = 14 which is an even number. Similarly, when we add 24 and 12, we will get 24 + 12 = 36 which is an even number.
- When we subtract two even numbers, the result is always an even number. For example, when we subtract 8 from 14 we will get 14 – 8 = 6. Similarly, when we subtract 24 from 36, we will get 36 – 24 = 12.
- When we add an even number and an odd number, the result is always an odd number. For example, if we add 3, which is an odd number and 6 which is an even number, we will get, 3 + 6 = 9 which is an odd number. Similarly, if we add 9 which is an odd number and 16 which is an even number we will get 9 + 16 = 25 which is an odd number.
- When we subtract an even number and an odd number, the result is always an odd number. For example, if we subtract 3, which is an odd number from 6 which is an even number, we will get, 6 – 3 = 3 which is an odd number. Similarly, if we subtract 12 which is an even number from 17 which is an odd number we will get 17 – 12 = 5 which is an odd number.
- When we multiply two even numbers, the result is always an even number. For example, if we multiply 6 which is an even number by 12 which is again an even number, the result will be 6 x 12 = 78 which is again an even number. Similarly, if we multiply 4 which is an even number by 8 which is again an even number, the result will be 4 x 8 = 32 which is again an even number.
- Zero is an even number. Zero is an even number because it is an integer multiple of 2, specifically 0 × 2.

**Representation of Sets of Odd Numbers and Even Numbers**

Now that we have learnt what we mean by odd numbers and even numbers is it possible for us to generalise them in a statement so as to represent them in set builder form? Let us find out.

In order to represent the odd numbers and even numbers in a generalised form, it is important to understand the condition which defines odd numbers and even numbers. We know that the basic condition for a number to be an even number is that it is divisible by 2. Therefore, if we want to generalise it and represent it in the set builder form, we will have,

**Set of Even Numbers = { x : x = 2k, where k is any integer }**

Similarly, we know that the basic condition for a number to be an odd number is that it is not divisible by 2. Also, we know that any number that is not divisible by 2 will leave a remainder 1 when divided by 2. Therefore, such a number should be of the form 2x + 1, where x is an integer. Therefore, if we want to generalise it and represent this in the set builder form, we will have,

**Set of Odd Numbers = { x : x = 2k + 1, where k is any integer }**

**Solved Examples**

**Example 1** Find the sum of

a) First 5 odd natural numbers.

b) First 5 even natural numbers

**Solution** We have been asked to find

a) First 5 odd natural numbers.

b) First 5 even natural numbers

Let us do them one by one.

In order to obtain the sum of first five odd natural numbers, we must first list down the first 5 odd natural numbers. We know that natural numbers start from 1 and go on 1, 2, 3, 4, 5 and on on.

Therefore, the first 5 odd natural numbers are 1, 3, 5, 7 and 9. The sum of these numbers will be 1 + 3 + 5 + 7 + 9 = 25

**Hence, the sum of first 5 odd natural numbers = 25**

Next we will find the sum of first 5 even natural numbers.

The first 5 even natural numbers are 2, 4, 6, 8 and 10.

The sum of these numbers will be 2 + 4 + 6 + 8 + 10 = 30

**Hence, the sum of first 5 odd natural numbers = 25**

**Example 2** List all odd numbers greater than 3 and smaller than 30.

**Solution** We have been asked to list all odd numbers greater than 3 and smaller than 30. Let us first list down all numbers that are greater than 3 and less than 30 in the table below –

4 | 5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 | 13 |

14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 |

24 | 25 | 26 | 27 | 28 |

29 | 30 |

Next, we will highlight the odd numbers in this table and mark them in yellow. We will get,

4 | 5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 | 13 |

14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 |

24 | 25 | 26 | 27 | 28 |

29 | 30 |

**Therefore, all odd numbers greater than 3 and smaller than 30 are 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 and 29.**

**Example 3**** **Using only once the digits 3, 4, 6, 7 from

- 3 – digit smallest even number.
- 4 – digit smallest even number with 6 at thousands place.
- 2 – digit odd number less than 40
- 4 – digit largest odd number with 3 at hundreds place.
- Smallest even number by using all the 4 digits.

**Solution** a) We have been given the digits – 3 , 4 6 and 7.

We need to find the 3 – digit smallest even number. **The smallest 3 digit even number will be 3 4 6.**

b) Again, we have been given the digits – 3 , 4 6 and 7

we need to find the 4 – digit smallest even number with 6 at thousands place.

**The 4 – digit smallest even number with 6 at thousands place is 6 3 7 4.**

c) We have been given the digits – 3 , 4 6 and 7.

We need to find a 2 – digit odd number less than 40. **The 2 – digit odd number less than 40 is 3 7.**

d) We have been given the digits – 3 , 4 6 and 7.

We need to find 4 – digit largest odd number with 3 at hundreds place. **The 4 – digit largest odd number with 3 at hundreds place will be 6 3 4 7**

e) We have been given the digits – 3 , 4 6 and 7.

We need to find the smallest even number by using all the 4 digits. **The smallest even number by using all the 4 digits will be 3 4 7 6**

**Key Facts and Summary**

- Even numbers are numbers that are divisible by 2, leaving a remainder 0.
- Odd numbers are numbers that are not divisible by 2, and always leave a remainder 1 when divided by 0.
- Set of Even Numbers = { x : x = 2k, where k is any integer }
- Set of Odd Numbers = { x : x = 2k + 1, where k is any integer }
- When we add two even numbers, the result is always an even number.
- When we subtract two even numbers, the result is always an even number.
- When we multiply two odd numbers, the result is always an odd number.
- When we multiply an even number and an odd number, the result is always an even number.
- When we add two even numbers, the result is always an even number.
- When we subtract two even numbers, the result is always an even number.
- When we add an even number and an odd number, the result is always an odd number.
- When we subtract an even number and an odd number, the result is always an odd number.
- When we multiply two even numbers, the result is always an even number.
- Zero is an even number.
- The numbers that have any of the five digits 1, 3, 5, 7 and 9 are all odd numbers.
- The numbers that have any of the five digits 0, 2, 4, 6 and 8 are all even numbers.