Even and Odd Numbers

Even numbers are divisible by 2 without a remainder. They always have the last digit as 0, 2, 4, 6, or 8. For example, 20, 22, 24, 26, 28, etc.

We mathematically express the set of even numbers as:
\( \{ x : x = 2k, \text{ where } k \in \mathbb{W} \} \)

Here, 
k: any integer 
W: the whole number set

On the other hand, odd numbers are numbers that cannot be evenly divided by 2. They have the last digit of 1, 3, 5, 7, or 9. For example, 21, 23, 25, 27, 29, etc.

We mathematically express the set of odd numbers as:
\( \{ x : x = 2k + 1, \text{ where } k \in \mathbb{Z} \} \)

Here, 
k: any integer
Z: the set of integers

There are certain differences between even and odd numbers that you should understand. We will now learn the key differences between these sets of numbers:

To find whether a number is even or odd, we follow different methods. Here are a few simple steps to determine them:


Step 1: Checking the one's place of the given number:
 

• A number ending in 0, 2, 4, 6, or 8 is even.
• A number ending in 1, 3, 5, 7, or 9 is odd. 

Step 2: Equal grouping of numbers:

• If you can group numbers into pairs without any left-out pairs, it is an even number.
• If one item remains unpaired after grouping, the number is odd.

The odd numbers and even numbers have various properties. Here’s a list of them:
 

• The sum of two even numbers is always an even number. 
  For example: \(2 + 4 = 6\)
   
• The sum of an even number and an odd number is always an odd number.
  For example: \(12 + 11 = 23\)
   
• The sum of two odd numbers is always an even number.
  For example:\( 3 + 5 = 8\)
   
• The difference between two even numbers is always an even number.
  For example: \(24 – 12 = 12\)
   
• The difference between an even number and an odd number is always an odd number.
  For example: \(20 – 9 = 11\)
   
• The difference between two odd numbers is always an even number.
  For example:\( 13 – 7 = 6\)
   
• The product of two even numbers is always an even number.
  For example: \(6 × 2 = 12\)
   
• The product of an even number and an odd number is always an even number.
  For example:\( 18 × 5 = 90\)

A visual representation of numbers in a chart helps students easily grasp them. Let’s look at the odd numbers from 1 to 100.

The even numbers from 1 to 100 are listed below:
 

Mastering even and odd numbers helps in quick calculations, pattern recognition, and solving mathematical problems efficiently.

• Remember that even numbers end with 0, 2, 4, 6, 8; odd numbers end with 1, 3, 5, 7, 9.
   
• Sum or difference of two even or two odd numbers is always even.
   
• Sum or difference of an even and an odd number is always odd.
   
• Multiplying any number by an even number gives an even result.
   
• Spot patterns in sequences to quickly identify even and odd numbers.

Even and odd numbers are basic concepts with practical uses beyond mathematics. Let’s look into some of them:

• Children often use even numbers to divide things equally. For example, 14 candies can be shared between two friends with no leftovers.
   
• In many buildings, rooms with odd numbers are placed on one side and even numbers on the other, making it easier to find specific rooms.
   
• In theaters and stadiums, seats are usually numbered in an odd-even pattern so visitors can quickly locate their seats.
   
• In sports, even numbers help create balanced teams, while an odd number of players usually means one person will be left out.
   
• When counting items like money, an odd number indicates that the total cannot be divided evenly.