Odd Numbers

Odd numbers are the numbers that cannot be arranged in pairs. The Ancient Greeks would consider numbers that could not be arranged in two rows as odd. This concept has changed over the millennia. For example, take any multiple of the number 2. You would realize that none of these numbers can be arranged in pairs of 2. Interestingly, all the integers except the multiples of 2 are odd numbers. You will learn about this property later in the article.

1.	What are Odd Numbers?
2.	List of Odd Numbers
3.	Properties of Odd Numbers
4.	Types of Odd Numbers
5.	FAQs on Odd Numbers

What are Odd Numbers?

Odd numbers definition is given as those numbers which cannot be divided into two parts equally.

Odd numbers are basically integer numbers that cannot be categorized into groups of two each. For example: 1, 3, 5, 7, etc. Let's visualize it using an example of footwear and cherries. Let us assume that we have footwear in counts of 1, 3, 5, and 7. On the other hand, we have cherries in the counts of 2, 4, 6, and 8. Look at the image given below in order to understand how the pairing of these numbers will work.

Odd and Even Numbers

It is to note here that the footwear, when odd in number, does not form a pair entirely. One among all remains unpaired. On the contrary, even numbers are those numbers that can be divided into two parts equally. For example: 2, 4, 6, 8, etc.

List of Odd Numbers

Let us have a look at the list of all the odd numbers from 1 to 200 and try to apply the knowledge we have learned here so far. Do note that none of the numbers given here are multiples of 2. You will also note that out of the first 200 numbers, only 100 numbers are odd numbers. Have a look at the list of the odd numbers from 1 to 200 given here.

1	3	5	7	9	11	13	15	17	19
21	23	25	27	29	31	33	35	37	39
41	43	45	47	49	51	53	55	57	59
61	63	65	67	69	71	73	75	77	79
81	83	85	87	89	91	93	95	97	99
101	103	105	107	109	111	113	115	117	119
121	123	125	127	129	131	133	135	137	139
141	143	145	147	149	151	153	155	157	159
161	163	165	167	169	171	173	175	177	179
181	183	185	187	189	191	193	195	197	199

The definition that we have learned above is applied in this table and it eases our work, doesn't it? Look carefully at the given table and try to notice some similarities between all these numbers given above.

Did you notice a pattern in the above odd numbers list?
In the odd numbers list, the one's place always remains 1, 3, 5, 7, or 9.

Properties of Odd Numbers

If you try to carry out a few BODMAS operations on the odd numbers, can you come to a common conclusion of all of the numbers? Well yes, there does exist a set of properties that applies not only for the odd numbers given in the list of 1 to 200 but are applicable to any odd number that you may come across. Given below is a list of the properties that will always apply for an odd number. Each of these properties can be explained in a detailed way as given below,

Addition of Two Odd Numbers: The addition of two odd numbers will always give an even number, i.e., the sum of two odd numbers is always an even number. For example, 3 (odd) + 5 (odd) = 8 (even)
Subtraction of Two Odd Numbers: Subtraction of two odd numbers will always give an even number. For example, 7 (odd) + 1 (odd) = 6 (even)
Multiplication of Two Odd numbers: Multiplication of two odd numbers will always give an odd number. For example, 3 (odd) × 7 (odd) = 21 (even)
Division of Two Odd Numbers: Division of two odd numbers will always give an odd number. For example, 33 (odd) ÷ 11 (odd) = 3 (odd)

Let's summarize our learning of properties using the table and simulation given below:

Arithmetic Operation	Result
Odd number + Odd number	Even number
Odd number - Odd number	Even number
Odd number × Odd number	Odd number
Odd number ÷ Odd number	Odd number

Types of Odd Numbers

Odd numbers are a list of all the numbers that are not the multiples of 2. So this seems like a vast set of numbers. So we can have many types of odd numbers starting from whether the odd numbers have factors or not, what is the difference between the two odd numbers, what is the position on the number line of the given odd numbers, and etc. Below given are the two main types of odd numbers.

Consecutive odd numbers

Let's say n is an odd number, then the number n and n + 2 are grouped under the category of consecutive odd numbers. They always have a difference of 2 between them and are consecutive in nature, hence the name consecutive odd numbers. For example 3 and 5, 11 and 13, 25 and 27, 37 and 39, 49 and 51, and so on. The list is never-ending.

Composite odd numbers

As the name suggests, Composite means made up of several parts or elements. These types of odd numbers are formed by the product of two smaller positive odd integers. The composite odd numbers from 1 to 100 are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, and 99.

Tips and Tricks on Odd Numbers

Given below is a list of a few tips and tricks on the topic of odd numbers. These will help you in remembering the concepts faster.

An easy method to differentiate whether a number is odd or even: divide it by 2
If the number is not divisible by 2 entirely, it'll leave a remainder 1, which indicates a number is an odd number and it can't be divided into 2 parts evenly.
If the number is divisible by 2 entirely, it'll leave a remainder 0, which indicates a number is an even number and it can be divided into 2 parts evenly.
Odd numbers always have 1, 3, 5, 7, or 9 in their unit's place. Even numbers always have 0, 2, 4, 6, or 8 in their unit's place.

Important Notes on Odd Numbers

Given below is a list of a few important notes on the topic of odd numbers. These will help you in understanding the concepts better.

Whole numbers, natural numbers, and integers can be categorized as odd numbers and even numbers.
Fractions and decimal numbers can never be categorized as odd numbers and even numbers.
When two numbers are divided, the numerator must be divisible by a denominator or the denominator must be a factor of the numerator, only then the output of division can be categorized as an odd number or even number, due to the limited applicability of decimal numbers.

Given below is the list of topics that are closely connected to Odd Numbers. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

Important Topics

What are Consecutive Numbers?

Number Systems

Example 1: Determine whether 135 is an odd number or not.

Solution:

We can find this out using the following steps: First, we should check the unit's place of the number. The unit place of the number is 5, which shows that number is an odd number. Now, we have to check the divisibility of the number by 2. On dividing the given number 135 by 2, we get the remainder as 1. This proves that 135 is an odd number. ∴ We determined that the number 135 is odd.
Example 2: Find the sum of the smallest 2 digit number and the largest 2 digit number and also prove that it is an odd number.

Solution:

We will follow the steps as mentioned. First, we will write down the smallest 2 digit number. The smallest 2 digit number is 10. Now, we write down the largest 2 digit number. The largest 2 digit number is 99. Now, we have to add the smallest 2 digit number and the largest 2 digit number. Hence, the sum of both the numbers gives 10 + 99 = 109. Now, check the unit's place of the number. The unit place of the number is 9, which shows that number is an odd number. Now we will check the divisibility of the number by 2. On dividing the given number 109 by 2, we get the remainder obtained is 1. This proves that the sum of the smallest 2 digit number and the largest 2 digit number is an odd number.

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FAQs on Odd Numbers

What are The Odd Numbers From 1 to 100?

The odd numbers from 1 to 100 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99.

What is an Even Number?

The numbers which can be divided into two parts evenly are called even numbers.

What are Two Consecutive Odd Numbers?

Two consecutive odd numbers are those numbers that are odd and have a difference of 2 between them. For example 3 and 5, 11 and 13.

Is 0 an Even or Odd Number?

0 is an even number as it is an integer multiple of 2.

Is Number 1 Odd or Even?

1 is an odd number. This is because it has the digit 1 at the unit's place. Also, if you try to divide 1 by 2, you will get a remainder as 1. Hence, 1 is an odd number.

What is The Formula of Odd Numbers?

The odd number formula to represent an odd number can be given as: Formula for odd numbers = 2n + 1, where, n ∈ Z (whole numbers).

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