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• Odd Numbers – Definition with Examples

Odd Numbers – Definition with Examples

A number is simply an arithmetic value used to represent quantity and make calculations. All numbers are formed from the numerals 0-9. Numbers exist as odd or even numbers and prime or composite numbers. Even numbers are all numbers that are divisible by 2 regardless of whether they are positive or negative integers. For example, 12 can be divided by 2 to give 6 without a remainder; -42 can also be divided by 2 to give -21 without a remainder. 

Prime numbers are the ones that have only two prime factors, i.e. 1 and the number itself. For example, 13 has no other factors except 13 and 1. And composite numbers are opposite of prime numbers since they have more than two factors. For instance, 6 is divisible by 1, 2, 3 and 6 itself.

In this article, we are going to focus on understanding odd numbers by determining their properties, the ways to identify them, and different forms of odd numbers.

What is Odd Number?

An odd number is a number that is not exactly divisible by two. This is an inclusive odd number definition because some books define it as a number that is not divisible by two. This is not entirely true since these numbers are divisible by two, but they have remainders. Remainder is the leftover number in a division problem, and it is always less than the divisor.

For example, 19 is not exactly divisible by two 192=

                        9

    Divisor- 2  ⟌19

                       18

                        1–remainder

This makes 19 an odd number.

Odd numbers cannot be divided into two equal parts. 

How to Identify Odd Numbers?

Odd numbers are identified using place values. Place value is the value of each digit in a number. The place values of numbers are read starting from right to left and are expressed as ones, tens, hundreds, thousands, and so on.

For instance, 1268.

Starting from right  8 – ones(1)

6 – tens(10)

 2 – hundreds(100)

1 – thousands(1000)

For odd numbers, we focus more on one’s place value. The first value of a number from the right side is the one’s place value. If it is not exactly divisible by two, then the number is an odd number. 

Here are some examples:

a) Is 15 an odd number?

The number at the one’s place value is 5. Five is not exactly divisible by 2; thus, 15 is an odd number.

b) Is 12427831 an odd number?

The numeral at the one’s place value is 1. Is one an odd number? Yes, since it cannot be exactly divided by two, the number is an odd number.

The technique of using place values is the simplest and the fastest to identify odd numbers. Otherwise, some people may use the long division method to determine if a number has a remainder or is exactly divisible by two.

Properties of Odd Numbers

Addition

When an odd number is added to another odd number, the resulting value is an even number.

Let’s take two odd numbers 3 and 9: 3+9=12, 12 is an even number.

When an odd number is added to an even number, the resulting number is an odd number. For example,  5 – odd number and 4 – even number, 4+5= 9, 9 is an odd number.

Subtraction

When an odd number is subtracted from an even number, the resulting number will be an odd number, like 10-3=7, 7 is an odd number.

When an odd number is subtracted from another odd number, the remainder is an even number.

5-3=2; so, is 2 an odd number? No, it is an even number.

When an even number is subtracted from an odd number, the remainder is an odd number, like in the examples 7- 4=3, 11- 6=5.

Multiplication

Multiplying an odd number with an odd number results in an odd number

7 x 3 = 21

5 x 7 = 35

Multiplying an even and odd number always results in an even number

7 x 2 = 14

6 x 5 = 30

Division

When we divide two odd numbers, the result will be an odd number. This is only true when the denominator is a factor of the numerator.

 9-numerator3- denominator = 3, denominator 3 is a factor of 9 where 33=9

If the denominator is an odd number but not a factor of the numerator, the resulting answer will not be a whole number but will have decimal points.

   95=1,8

Types of odd numbers

There are two types of odd numbers, namely composite odd numbers and consecutive odd numbers.

Composite odd numbers

Composite numbers are positive integers with more than two factors, including 1 and the number itself. When it comes to composite odd numbers, they are positive integers which cannot be divided by 2 but have more than two factors.

For example, 9 has 3, 1, and 9 as its factors; 15 has 3,5,1, and15 as its factors.

Other examples are 9, 15, 21, 25, 27, 33, 35…

They are formed by the product of two smaller positive integers.

Consecutive odd numbers

Consecutive odd numbers as the name suggests are odd numbers that follow each other in a sequential order with a difference of two. 

If x represents an odd number, then x and x+ 2 is a consecutive numbers x=3

x+2=?,  3+2=5, so, 3 and 5 are consecutive odd numbers.

The examples are 3, 5, 7, 9, 11, 13 … and so on.

The difference between the list of odd numbers given is 2; hence, these numbers are consecutive odd numbers.

Odd numbers can also be odd prime numbers or odd natural numbers.

• Odd prime numbers. These are prime numbers which are odd numbers and have only 1 and the number itself as the factors. Surprisingly, all prime numbers except 2 are odd numbers. For example, 3, 5, 7, 11,13 …
• Odd natural numbers. These are all positive counting numbers that are odd starting from 1 to infinity, such as 1, 3, 5, …

Odd number chart (1 to 100)

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

List of odd numbers from 1 to 10

1, 3, 5, 7, 9

 Is zero an odd number? 

 Zero is not an odd number. It is actually an even number.

What is the smallest odd composite number?

From the definition of a composite odd number, 9 qualifies to be the smallest composite odd number. This is because all the other odd numbers before 9, such as 1, 3, 5, and 7, have only 1 and the number itself as the factors, while 9 has 3, 1, and 9 as the factors.

General form of odd numbers

The general form of integers is

 n = 2k + 1 or n = 2k – 1, where n is the odd number and k is any positive integer.

For n = 2k + 1, the integers range from 0, 1, 2, 3 to infinity, while for n = 2k- 1, the value of integers range from 0, 1, 2, 3, 4 to infinity.

Solved examples on odd numbers

Example 1

Determine if the following numbers are odd numbers:

9873457

The value at the one’s place value is 7. Since 7 is not exactly divisible by 2, then the number is an odd number.

790450

The integer at the one’s place value is 0. The number is not odd since 0 is an even number.

Example 2

Is 9 an odd number?

92= 4 remainder 1

9 is an odd number

Is 1 an odd number?

12= 0 remainder 1

1 is an odd number

Example 3

List out all the odd numbers in the following list of numbers:

1, 23, 22, 67, 32, 57, 90, 89, 64, 103, 780, 10007, 9, 33, 71

 Answer: 1, 23, 67, 57, 89, 103, 10007, 9, 33, 71.

Conclusion

A number can never be an odd and even number at the same time. It can only be either even or odd. It is important to note that 1 is the first positive integer since most people rule it out and start from 3. If you add all the odd numbers from 1 to any other odd number, the resulting sum will always be a perfect square. For example, 1+ 3+ 5+ 7=16, 16 is a perfect square(4 4). To understand odd numbers, one should know its properties and types.

Frequently asked questions on odd numbers

What is an odd number?

An odd number is a number that is not exactly divisible by two.

What are the first 10 odd numbers?  

They are 1, 3, 5, 7, and 9.

How do you determine if a number is odd or even?

You can determine it by dividing the number by two. If the number is exactly divisible by two, it is an even number; otherwise, it is an odd number.

What are the odd numbers from 1 to 100?

Odd numbers are all numbers with a 1, 3, 5, 7, and 9 at the one’s place value. For example, 11, 23, 35, 47, and 59.

What is the difference between an odd and even number?

An odd number is not exactly divisible by two since it has a remainder, while even numbers are exactly divisible by two. For instance, 13 and 42. 132= 6 remainder 1, then, 13 is an odd number. 422 =12, then, 42 is an even number since there is no remainder.