# Even and Odd Numbers

### What are Even numbers?

- Numbers having
**0, 2, 4, 6, or 8 at one’s place**are called even numbers. - Even numbers are
**divisible by 2 without any leftovers**(remainders)**Examples:**6, 10, 22, 44, 96, 118, 792, etc.

### What are Odd numbers?

- Numbers having 1, 3, 5, 7, or 9 at one’s place are called odd numbers.
- Odd numbers have a leftover when divided into two equal groups.
**Examples**: 3, 11, 27, 71, 99, 111, 797, etc.

Regardless of how many digits a number has, you can identify if it is odd or even by looking at the last digit. For example, the numbers 8, 14, and 7,950 are even because their one’s place numbers are 8, 4, and 0.

Similarly, the numbers 63, 1,727, and 125 are odd because they have 3, 7, and 5 at their one’s place.

### Arithmetic Rules for Odd and Even Numbers:

- The sum of two even numbers is always an even number.
**Even + Even = Even**

Example: 16 + 12 = 28 - The sum of two odd numbers is always an even number.
**Odd + Odd = Even**

Example: 13 + 17 = 30 - The sum of an odd number and an even number is an odd number.
**Even + Odd = Odd**

Example: 24 + 35 = 59 - The difference of two even numbers is an even number.
**Even – Even = Even**

Example: 14 – 12 = 2 - The difference of two odd numbers is an even number.
**Odd – Odd = Even**

Example: 17 – 5 = 12 - The difference between an even number and an odd number is an odd number.
**Even – Odd = Odd****Odd – Even = Odd**Examples: 30 – 13 = 17, 25 – 20 = 5 - The product of two even numbers is always an even number.
**Even x Even = Even**

Example: 8 x 6 = 48 - The product of an even number and an odd number is always an even number.
**Even x Odd = Even**

Example: 2 x 7 = 14 - The product of two odd numbers is always an odd number.
**Odd x Odd = Odd**

Example: 9 x 3 = 27 - An even number can be represented as
**2n**, where n is an integer.

Example: 2(4) = 8 - When an even number is divided by 2, there is no remainder.

Example: 24 ÷ 2 = 12 - An odd number can be represented as
**2n + 1**, where n is an integer.

Example: 2(2) + 1 = 5 - When an odd number is divided by 2, it leaves a remainder of 1.

Example: 17 ÷ 2 = 8 remainder 1

These properties help us understand the characteristics and behavior of odd and even numbers.

## Solved Examples of Even and Odd Numbers

**Example 1.**

Is 15 + 31 even or odd?

Solution :

15 = odd number

31 = odd number

**Rule : odd + odd = even**

15 + 31 = 46, an even number

So, 15 + 31 is even.

**Example 2 :**

Is 16 + 110 even or odd?

Solution :

16 = even number

110 = even number

**Rule : even + even = even**

Moreover,

16 + 110 = 126, an even number

So, 16 + 110 is even.

**Example 3 :**

Is 9 + 88 even or odd?

Solution :

9 = odd number

88 = even number

**Rule : odd + even = odd**

Moreover,

9 + 88 = 97, an odd number

So, 9 + 88 is odd.

**Example 4 :**

Is 200 + 111 even or odd?

Solution :

200 = even number

111 = odd number

**Rule : even + odd = odd**

Moreover :

200 + 111 = 311, an odd number

So, 200 + 111 is odd.

**Example 5 :**

Is 100 x 52 even or odd?

Solution :

100 = even number

52 = even number

**Rule : even x even = even**

Moreover,

100 x 52 = 5200, an even number

So, 100 x 52 is even.

**Example 6 :**

Is 11 x 12 even or odd?

Solution :

11 = odd number

12 = even number

**Rule : odd x even = even**

Moreover,

11 x 12 = 132, an even number

So, 11 x 12 is even.

**Example 7 :**

Is 89 x 17 even or odd?

Solution :

89 = odd number

17 = odd number

**Rule : odd x odd = odd**

Moreover,

89 x 17 = 1513, an odd number.

So, 89 x 17 is odd.

**Example 8:**

Is 76 x 13 even or odd?

Solution :

76 = even number

13 = odd number

**Rule : even x odd = even**

Moreover,

76 x 13 = 988, an even number.

So, 76 x 13 is even.