# Odd Numbers

Numbers or set of integers on a number line can either be odd or even.

What is an Odd Number?

A number of the form 2k+1, where k∈Z  (i.e. Integers)

An odd number is an integer which is not a multiple of 2..

If these numbers are divided by 2, the result or remainder should be a fraction or 1.

In the number line, 1 is the first positive odd number.

Odd Number List

-5, -3, -1, 1, 3, 5 , 7 , 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31 etc.

Even Number List

-6, -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 etc.

Consecutive Odd Numbers

If ‘a’ is an odd number, then ‘a’ and ‘a + 2’ are called consecutive odd numbers.

For example, 15 and 17 are consecutive odd numbers and as are 29 and 31.

Properties of Odd numbers-

1. Adding two odd numbers- Odd + Odd = Even

Let’s prove-

Let two odd number be a and b.

These numbers can  be written in the form where

$a = 2k_{1}+1$

and $b = 2k_{2}+1$

where $k_{1}, k_{2}\in \mathbb{Z}$

$(2k_{1}+1)+(2k_{2}+1)$

$2k_{1}+2k_{2}+2$

$2(k_{1}+k_{2}+1)$ which is surely divisible by 2.

 Lets Work Out- Example- Find the sum of the smallest and the largest 3 digit odd number and also prove that it is divisible by 2. Solution- Smallest 3 digit odd number = 101 Largest 3 digit odd number = 999 Sum of both the numbers = 101 + 999 = 1100 The number 1100 is divisible by 2(as per divisibility rule of 2). This proves that the number is even.
1. Subtracting two odd numbers- Odd – Odd = Even

You can prove the result similarly as proved for addition of two odd numbers.

1. Multiplication of two odd numbers- Odd $\times$ Odd  = Odd

Let’s prove-

Let two odd number be a and b.

These numbers can  be written in the form where

$a = 2k_{1}+1$

and $b = 2k_{2}+1$

where $k_{1}, k_{2}\in \mathbb{Z}$

$a \times b = (2k_{1}+1)(2k_{2}+1)$

$a \times b = 4k_{1}k_{2}+2k_{1}+2k_{2}+1$

The above equation can be re-written as $a \times b = 2(2k_{1}k_{2}+k_{1}+k_{2})+1 = 2(x)+1$

Thus the multiplication of two odd number results in odd number.

1. Division of two odd numbers- $\frac{Odd}{Odd} = Odd$

Division of two odd numbers always result in Odd number if and only if the denominator is a factor of numerator, or else the number result in decimal point number.

Composite Odd Number

A composite odd number is a positive odd integer which is formed by multiplying two smaller positive integers or multiplying the number with one. The composite odd numbers up 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, 99.