## Odd Numbers

**Odd Numbers:** When we talk about even and odd numbers the simplest way we use to find out whether the given number is even or odd is by checking the divisibility of that number by 2. What does it mean? It simply means that if a number is completely divisible by 2, without leaving any remainder, the given number is then an even number. When the given number is not completely divisible by 2 and leaves any remainder then the number is considered as an odd number. Generally, odd numbers cannot be arranged in pairs.

Look at these examples; 9, 25, and 14.

Firstly we will check the divisibility of the given number by 2.

So, 9 ÷ 2 =4.5, 25 ÷ 2= 12.5, and 14 ÷2 =7

As you can see that 9 and 25 are not completely divided by 2 as their quotient has a decimal number which means, they are left with some remainders here in 9 and 25 the remainder is 1, and if you observe the number 14 which is completely divisible by 2 leaving no remainder. So, 9 and 25 are odd numbers. Let’s learn more about odd numbers, definitions, properties, and important points

## Odd Numbers Definition

**Odd numbers are numbers that cannot be divided by 2 evenly. An odd number is a number that is not divisible by 2 and is left with some remainder or a number that can not be divided into two equal groups or pairs. An odd number is written in the form n=2k+1, where k is an integer.**

Now, Let’s understand this form of representing an odd number in more detail.

If we put any value of k, in **n=2k+1, ** then the value of n is the odd number.

For example: If we take k as 3, and put it into **n=2k+1,**

We will get n = 2 x 3 + 1 = 7, which is an odd number as 7 is not divisible by 2.

If we take k as 0, and put it into **n=2k+1,**

We will get n = 2 x 0 + 1 = 1, which is an odd number as 1 is not divisible by 2, and 1 is the smallest odd number.

## Identification of Odd Numbers

We have many different methods to identify an odd number in a more easy and simple way. Some of the methods are listed below as the easiest way to identify whether a given number is even or not.

- Check the divisibility of a number by 2, if it is not completely divisible then it is an odd number.
- If the last digit of a number ends up with 1, 3, 5, 7, or 9. It is odd. Otherwise, If the last digit of a number ends up with 0, 2, 4, 6, or 8. It is even.
- The odd number is written in the form
**n=2k+1**, where k is an integer.

For example,

check if 39 is an odd number or not by all three methods of identification.

If we go with the first method, 39 ÷ 2= 19 as a quotient and 1 as a remainder. So, by this method 39 is an odd number.

Now, from the second method, the last digit of 39, or 39 ends with 9, So by this method also, 39 is an odd number.

Now, by the third method, an odd number is written in the form **n=2k+1, **where k must be an integer. Here, we have the value n=39,

**n=2k+1,**

39=2k+1

39-1=2k

38=2k

So, here we got the value of k=19,

Again putting the value of k=19, it into the form **n=2k+1, then**

n=2k+1, n=2x19+1, n=39

These are the three methods by which we will find out whether the number is odd or even. In many cases, the easiest method is looking for the last digit or one place of the given number if it contains 1, 3, 5, 7, or 9. It is odd.

**◙ **873 is an odd number as the last digit of 873 is 3.

**◙ **8457 is also an odd number as the last digit of 8457 is 7.

**◙ **241589 is an odd number as the last digit of 241589 is 9,

**◙ **587428 is also not an odd number as the last digit of 587428 is 8, and so it is an even number.

## Odd Numbers 1 to 100

List of odd Numbers 1 to 100 | ||||

1 | 21 | 41 | 61 | 81 |

3 | 23 | 43 | 63 | 83 |

5 | 25 | 45 | 65 | 85 |

7 | 27 | 47 | 67 | 87 |

9 | 29 | 49 | 69 | 89 |

11 | 31 | 51 | 71 | 91 |

13 | 33 | 53 | 73 | 93 |

15 | 35 | 55 | 75 | 95 |

17 | 37 | 57 | 77 | 97 |

19 | 39 | 59 | 79 | 99 |

## Odd Numbers up to 200

List of odd Numbers 1 to 200 | |||||||||

1 | 21 | 41 | 61 | 81 | 101 | 121 | 141 | 161 | 181 |

3 | 23 | 43 | 63 | 83 | 103 | 123 | 143 | 163 | 183 |

5 | 25 | 45 | 65 | 85 | 105 | 125 | 145 | 165 | 185 |

7 | 27 | 47 | 67 | 87 | 107 | 127 | 147 | 167 | 187 |

9 | 29 | 49 | 69 | 89 | 109 | 129 | 149 | 169 | 189 |

11 | 31 | 51 | 71 | 91 | 111 | 131 | 151 | 141 | 191 |

13 | 33 | 53 | 73 | 93 | 113 | 133 | 153 | 173 | 193 |

15 | 35 | 55 | 75 | 95 | 115 | 135 | 155 | 175 | 195 |

17 | 37 | 57 | 77 | 97 | 117 | 137 | 157 | 177 | 197 |

19 | 39 | 59 | 79 | 99 | 119 | 139 | 159 | 179 | 199 |

## Properties of Odd Numbers

**1. Properties of Addition:**

- Adding two odd numbers will always have an even number in the final result.

For example, 9+11 =20, 15+25=40, 23+1=24

- The sum of one odd number and one even has always resulted in an odd.

For example, 7+10 =17, 87+2=89, 30+9=39

**2. Properties of Subtraction:**

- The difference between two odd numbers is always an even number.

For example, 13-3=10, 27-9=18, 47-11=36

- The difference between one odd number and one even is always an odd number.

For example, 27-10=17, 80-5=75, 98-7=91

**3. Properties of Multiplication.**

- The multiplication of two odd numbers always results in an odd number.

For example, 7 X 3=21, 11 X 3=33, 9 X 9=81

- The multiplication of one odd number and one even number has the results as an even number.

For example, 3 X 14=42, 16 X 5=80, 22 X 5=110

### Summary of Properties of Even Numbers

Properties of Addition: | ||

1. | Odd + Odd | Even |

2. | Odd + Even | Odd |

Properties of Subtraction: | ||

1. | Odd - Odd | Even |

2. | Even - Odd | Odd |

Properties of Multiplication | ||

1. | Odd X Odd | Odd |

2. | Even X Odd | Even |

## Types of Odd Numbers

Odd numbers can be categorized into two types on the basis of the characteristics of a whole number.

**1. Composite odd numbers:** Those odd numbers which cannot be divided by 2 and that have a factor other than 1 and itself, are known as composite odd numbers. Basically, the number must fulfill both the criteria at the same time i.e for the odd number and a composite number also. For example,

9, 15, 21, 25, 27, 33, 45, 49.

**2. Consecutive odd numbers**: In counting, every alternate number is an odd number. Consecutive odd numbers are the odd numbers that appear at a difference of 2 between each odd number. Let’s suppose ‘*n’* is an odd number, then the numbers ‘*n’* and ‘*n+2’* are consecutive odd numbers. For example, if n= 3 then *n+2 i.e *3 + 2=5, hence, 3 and 5 are consecutive odd numbers.

The consecutive odd numbers from 1 to 20 are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19.

### Important points on Odd Numbers

- 1 is the smallest odd number.
- Any Fractions and Decimals are not odd numbers, because they are not whole numbers.
- Counting from 1, Every alternate number is an odd number, that alternate with an even number.
- The addition of all the odd numbers from 1 to any given number I.e the sum of n odd numbers is calculated by S
*n*=**n².** - The sum of all the odd numbers from 1 to 200 is 10,000.
- Odd numbers are the opposite of even numbers.

## Odd Numbers- Solved Questions

**Q1: Check whether 5979 is odd or not.**

**Answer: **Checking the unit place of the given 5979 that ends with 9, which is an odd number. So, 5979 is an odd number.

**Q2: Find the sum of the odd numbers between 1 to 60?**

**Answer:** Every alternate number in counting is an odd number. So, there are 30 odd numbers present between 1 to 60.

hence, n = 30

We come to know that the sum of odd numbers is **n²**

I.e Sn = **n²**

So, S*n*= (30)**² **= 900

So, the sum of the odd numbers between 1 to 60 is 900, which is a perfect square.

**Q3: Find out all the odd numbers from the given set of numbers 666, 383, 945, 7852, 1111, 46521, 65787, 48572, 1894255, 223415.**

**Answer: **To easily differentiate an odd number from a given set of numbers, we will only consider the last digit of the number or the number in the unit place. Hence, The odd numbers from the given set of numbers are 383, 945, 1111, 46521, 65787, 1894255, and 223415.

**Q4: Find the sum of odd numbers between 1 and 30?**

**Answer:** The odd numbers between 1 and 30 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, and 29. The total number of odd numbers between 1 and 30 is 15,

hence, n = 15

We come to know that the sum of odd numbers is **n²**

So, S*n*= (15)**² **= 225

So, the sum of the odd numbers between 1 and 30 is 225.

**Q5: What is the sum of the first eight consecutive odd numbers between 40 to 60.**

**Answer: **First, we have to find the eight consecutive odd numbers between 40 to 60, and these are

41, 43, 45, 47, 49, 51, 53, and 55. So, the sum of the first eight consecutive even numbers between 40 to 60 is 41 + 43 + 45 + 47 + 49 + 51 + 53 + 55 = 384

So, the sum of the first eight consecutive odd numbers between 40 to 60 is 384.

**Q6: Determine whether 547 is an odd or even?**

**Answer:** We can find out an odd number using any of the given two methods:

First Method: We can also check the divisibility of the given number by 2. As the given number 547 is not completely divisible by 2, here, we get the remainder as 1. So, the given number is an odd number.

Second Method: Checking the units place digit of the number. The unit place of the number is 7, which shows that number is an odd number.

**Q7: Choose the correct answer.**

**The Sum of two odd numbers is always an even number.****The Difference between two odd numbers is always an odd number****Both statements are correct.****None of the above is correct.**

**Answer: **The correct answer is **option a).** As Odd + Odd = is always an even.

The statement in option **a) ** is correct while Odd - Odd is also an even, which is incorrectly described in the statement of option **b)**

**Q8: The results of which of the following numbers are odd?**

**18 × 19****23 × 39****77 × 60****64 × 46**

**Answer:** The correct answer is option **b). **As we know that the multiplication of two odd numbers is always an odd number, we have to find out the two odd numbers multiplication in all the above options.

**Q9: Find the sum of the smallest 2-digit odd number and the smallest 3-digit odd number?**

**Answer: **The smallest two-digit odd number is 11 and The smallest three-digit odd number is 101. So 101 + 11 =112 Hence, the sum of the smallest 2-digit odd number and the smallest 3-digit odd number is 112.

**Q10: Find the first five composite odd numbers** **between 1 to 30?**

**Answer: **The composite odd numbers are numbers that cannot be divided by 2 and that have a factor other than 1 and itself. So, the first five composite odd numbers are 9, 15, 21, 25, and 27.