Odd Numbers

Introduction

Odd Numbers and even numbers are one of the many types of classification of numbers. The odd numbers are not even, and the numbers which are even are not odd. Both are inverse to each other. Odd and even numbers are differentiated by whether the number is divisible by 2 or the number is a multiple of 2. An odd number, when divided by two, leaves a remainder of 1. On the other side, an even number, when divided by 2, leaves a remainder of 0. Even numbers are divisible by two and multiples of 2.

• Example − 1, 3, 5, etc… are odd numbers, and 0, 2, 4, etc… are even numbers.

Number system

A Number system is a system of writing or expressing numbers in various ways or dividing the numbers into different types.

There are various types of number systems

If we divide the numbers using the base of the number −

• Binary Number System (base 2)

• Octal Number System (base 8)

• Decimal Number System (base 10)

• Hexadecimal Number System (base 16)

If we divide the numbers whether they are divisible by 2:

• Odd numbers (not divisible by 2)

• Even Numbers (divisible by 2)

If we divide the numbers depending on the number of factors, that is whether it has factors other than 1 and itself −

• Prime Numbers (No factors other than one and itself)

• Composite Numbers (factors other than one and itself)

There are various other kinds of number systems that divide a set of numbers based on the respective properties.

Even and Odd numbers

Odd Numbers

If a number is not divided by 2, then it is an odd number. It is not a multiple of 2. It is in the form of $\mathrm{(2\times\:n)\:+\:1\:,\:where\:n\:\varepsilon\:Z}$. If a number is not odd, then it is an even number. The first positive odd number is 1.

• Example − 1, 13, 25, 37, etc…

Even Numbers

If a number is divided by 2 then it is an even number. It is a multiple of 2. It is in the form of $\mathrm{2\times\:n\:,\:where\:n\:\varepsilon\:Z}$ .If a number is not even, then it is an odd number.

• Example − 10, 22, 34, 46, etc…

Properties of Odd numbers

There are few properties regarding arithmetic operations of odd numbers.

Addition of two odd numbers − Adding two odd numbers gives an even number.

Take two odd numbers $\mathrm{a\:=\:2\times\:n\:+\:1\:,\:b\:=\:2\times\:m\:+\:1\:,\:Where\:n\:,m\:\varepsilon\:Z}$

Subtracting 𝑎 and 𝑏 − $\mathrm{\:\:\:\:\:a\:-\:b\:=\:2\times\:n\:+\:1\:-\:(2\times\:m\:+\:1)}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:a\:-\:b\:=\:2\times\:(n\:-\:m)\:=\:2\times\:(n\:-\:m)}$

$\mathrm{a\:-\:b}$ is in the form of an even number. So, the subtraction of two odd numbers results in an even number.

• Example − 25 - 13 = 12, 33 - 13 = 20

Multiplication of two odd numbers: Multiplication of two odd numbers will give an odd number.

Take two odd numbers $\mathrm{a\:=\:2\times\:n\:+\:1\:,\:b\:=\:2\:2\times\:m\:+\:1\:,\:where\:n\:,\:m\:\varepsilon\:Z}$

Multiplying 𝑎 and 𝑏 − $\mathrm{\:\:\:\:\:a\:\times\:b\:=\:2\:\times\:n\:+\:1\:\times\:(2\:\times\:m\:+\:1)}$

$\mathrm{\:\:\:\:\:a\:\times\:b\:=\:4\:\times\:n\:\times\:m\:+\:2\:\times\:n\:+\:2\:\times\:m\:+\:1\:=\:2\:\times\:(2\:\times\:n\:\times\:m\:+\:n\:+\:m)\:+\:1}$

$\mathrm{\:\:\:\:\:\:\:\:a\:\times\:b\:=\:2\:\times\:(2\:\times\:n\:\times\:m\:+\:n\:+\:m)\:+\:1\:=\:2\:\times\:l\:+\:1\:\:\:\:where\:l\:=\:2\:\times\:n\:\times\:m\:+\:n\:+\:m}$

$\mathrm{a\:\times\:b}$ is in the form of an odd number. So, the multiplication of two odd numbers results in an odd number.

• Example − $\mathrm{3\:\times\:5\:=\:15\:,\:13\:\times\:3\:=\:39}$

Division of two odd numbers − Division of two odd numbers will result in an odd number. If the numerator is divisible by the denominator otherwise it would result in a decimal number.

Take two odd numbers $\mathrm{a\:=\:2\times\:n\:+\:1\:,\:b\:=\:2\:\times\:m\:+\:1\:,\:\:\:\:where\:n\:,\:m\:\varepsilon\:Z}$

Dividing 𝑎 by 𝑏 − $\mathrm{\frac{a}{b}\:=\:\frac{2\:\times\:n\:+\:1}{2\:\times\:m\:+\:1}}$ the numerator should be divisible by the denominator,means the product of the denominator and some other number, let’s say 𝑐 is equal to the numerator. The denominator is an odd number, and the numerator is also odd. Only the product of two odd numbers will give an odd number which means 𝑐 should also be odd. Therefore, the division of two odd numbers will result in an odd number.

• Example − $\mathrm{\frac{15}{5}\:=\:3\:,\:\frac{45}{3}\:=\:15}$

Arithmetic Operation	Result
Addition (Odd + Odd)	Even Number
Subtraction (Odd - Odd)	Even Number
Multiplication (Odd × Odd)	Odd Number
Division (Odd / Odd) Given that numerator is divisible by the denominator.	Odd Number

Types of Odd Numbers

Consecutive Odd numbers

If two odd numbers are called consecutive odd numbers then the difference between them is equal to 2.

• Example − $\mathrm{(3\:,\:5)\:,\:(11\:,\:13)\:,\:(29\:,\:31)}$ are three pairs of consecutive odd numbers.

Composite Odd numbers

The numbers which are both composite (factors other than one and itself) and odd are called composite odd numbers.

• Example − 9, 15, 21, 25, etc…

Solved examples

1) Find the sum of the smallest odd number and the largest two-digit odd number?

Answer − The smallest odd number = 1

The largest two-digit odd number = 99

$\mathrm{Sum\:=\:1\:+\:99\:=\:100}$

2) Write the result of the following operations that the number is odd or even?

$\mathrm{25\:+\:37\:=\:?\:,\:29\:-\:17\:=\:?\:,\:37\:\times\:17\:=\:?\:,\:\frac{49}{7}\:=\:?}$

Answer − 62 is Even, 12 is even, 629 is Odd, 7 is Odd.

Conclusion

In this tutorial, we learned about the number system, various kinds of the number system, even and odd numbers, properties of odd numbers, consecutive odd numbers, composite odd numbers, and a few examples.

FAQs

1. What is the largest and smallest five-digit odd number?

10001 → is the smallest five-digit odd number.

99999 → is the largest five-digit odd number.

2. Check whether the numbers, 3672, 456789, 98765, and 138269 are odd or even?

The numbers 456789, 98765, and 138269 are odd because they are not divisible by 2.

The number 3672 is even because it is divisible by 2.

3. How do we know that a number is composite odd?

If the number is both composite (factors other than one and itself) and odd, then the number is a composite odd number

4. What is the difference between two consecutive odd numbers?

The difference between two consecutive odd numbers is equal to 2.

5. What is the smallest even number?

Zero is the smallest even number because when divided by 2 it gives both quotient and remainder equal to zero. So, zero is an even number, and it is the smallest even number.