# Odd Numbers

Odd numbers or integers are part of whole numbers that are partially divisible into pairs. Thus all numbers except the multiples of 2 are odd numbers. They are in the form of 2k+1, where k ∈ Z (integers) are called odd numbers.

Some examples are 1, 3, 5, 7, and so on. They are just the opposite of even numbers. Odd numbers can be represented on a number line:

Looking at the examples, we can consider whether all odd numbers are prime.

No, it is not. However, all prime numbers are odd except 2.

## List of Odd Numbers

Practice writing the odd numbers from 1 to 1000.

## Properties

The 4 main properties are:

Any 2 odd numbers, when added, always gives an even number.

Odd number + Odd number = Even number

Proof:

Let a and b are 2 odd numbers

They are written in the form

a = 2k1 + 1, b = 2k2 + 1 where k1, k2 ∈ Z

Adding a and b, we get

(2k1 + 1) + (2k2 + 1)

=> 2k1 + 2k2 + 2

=> 2(k1 + k2 + 1)

This expression is divisible by 2s

Similarly, the sum of the first n odd numbers follow Sn = n2 rule.

### Subtracting 2 Odd Numbers

When an odd number is subtracted from the other, it always gives an even number. It is similar to adding two odd numbers.

Odd Number – Odd number = Even number

### Multiplication of 2 Odd Numbers

When an odd number is multiplied by another odd number, the product is also an odd number.

Odd number × Odd number = Odd number

Proof:

Let a and b are 2 odd numbers

a = 2k1 + 1, b = 2k2 + 1 where k1, k2 ∈ Z

Now, a × b = (2k1 + 1)(2k2 + 1)

=>4kk2 + 2k1 + 2k2 + 1

=>2(2k1 k2 + k1 + k2) + 1

This expression is an odd number

### Division of 2 Odd Numbers

Division of 2 odd numbers always results in an odd number when the denominator is a factor of the numerator.

Odd number ÷ Odd number = Odd number

## Types

There are 2 types of odd numbers. They are:

### Consecutive Odd Numbers

They are 2 odd numbers that come one after the other in a sequence. If ‘a’ is an odd number, then the consecutive odd number corresponding to a is ‘a + 2’.They can be positive or negative.

Examples

Positive Consecutive Odd Number

• 5 and 7
• 11 and 13
• 51 and 53
• 101 and 103

Negative Consecutive Odd Number

• -5 and -7
• -11 and -13
• -51 and -53

### Composite Odd Number

They are positive odd numbers obtained by multiplying 2 smaller positive integers or multiplying the number with 1.

Given are the composite odd numbers till 100:

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.

Thus, the smallest odd composite number is 9.

## Solved Examples

Determine if 145 is an odd number or not.

Solution:

The given number is 145
Checking the divisibility of the number by 2, we get the remainder by 1, which proves that 145 is an odd number.

Simplify:
a) (-5) + (-9)
b) (-11) – (-3)
c) (-3) × (-7)
d) (-9) ÷ (-3)

Solution:

a) (-5) + (-9) = -14
b)(-11) – (-3) = -8
c) (-3) × (-7) = 21
d) (-9) ÷ (-3) = 3

The sum of three consecutive odd numbers is 51. Find the numbers.

Solution:

Let x be an odd number
Then the next consecutive odd number are x + 2 and the next term is x + 2 + 2 = x + 4
Now,
x + (x + 2) + (x + 4) = 3x + 6 = 51
=> 3x = 51 – 6
=> x = ${\dfrac{45}{3}}$ = 15
Hence, the other numbers are 15 + 2 = 17, 15 + 4 = 19
Thus the 3 consecutive odd numbers are 15, 17, and 19.

The sum of 5 consecutive odd numbers is 145. Find the third number in the sequence.

Solution:

Let x be an odd number
Then the next consecutive odd numbers are:
x + 2
x + 2 + 2 = x + 4
x + 2 + 2 + 2 = x + 6
x + 2 + 2 + 2 + 2 = x + 8
Now,
x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 5x + 20 = 145
=> 5x = 145 – 20
=> 5x = 125
=> x = ${\dfrac{125}{5}}$ = 25
Hence, the other numbers are
25 + 2 = 27
27 + 2 = 29
29 + 2 = 31
31 + 2 = 33
Thus, the third number in the sequence is 29.