Whole numbers are divided into even and odd numbers in mathematics. Odd and even numbers are both distinct sets of numbers, and no single number can be both even and odd. A number can be either even or odd. Even numbers are those that can be divided evenly by two, whereas odd numbers cannot be divided evenly by two.

Even and odd numbers:-

A number that can be divided into two equal groups is known an even number. An odd number, on the other hand, is one that cannot be divided into two equal groups.

Even and odd numbers are also referred  as:

Even Number: An even number is one that is divided by two and leaves 0 as the remainder.

Odd Number: An odd number is one that is not divisible by two and leaves one as the remainder.

You always look for the last digit to distinguish between odd and even numbers. An even number’s last digit is always 0, 2, 4, 6, or 8, while an odd number’s last digit is always 1, 3, 5, 7, or 9.

Examples of even and odd numbers:-

Even numbers include the following numbers: -22, -10, 0, 6, 18, 234.

Because they end with 0, 2, 4, 6, or 8, the numbers above are even.

odd numbers include the following numbers:-

-101, -17, 1, 9, 23, 985 .

Because they end with 1, 3, 5, 7, or 9,the numbers above are odd.

Set Representation of Even and Odd Numbers

X: x = 2k where k W, where k is an integer and ‘W’ is the set of whole numbers, represents the set of even integers.

X: x = 2k – 1 where k N, where ‘k’ is an integer and ‘N’ is the set of natural numbers, represents the set of odd numbers.

Properties of  Even and Odd Numbers:-

When it comes to algebraic operations, odd and even integers have unique qualities (addition, subtraction, and multiplication). We always receive an even or odd number when we apply algebraic operations to two even or odd numbers. We don’t include division since when discussing unique qualities, division might occasionally result in fractions.

• When two even integers are added or subtracted, the outcome is always an even number. For instance, 6 + 4 = 10

      2 = 6 – 4

• When an even number and an odd number are added or subtracted, the outcome is always odd.

     Like , 7 + 4 = 11

      3 = 7 – 4

• When two odd integers are added or subtracted, the outcome is always an even number.

 For instance, 7 + 3 = 10.

    7- 3 =4

• When two even numbers are multiplied, the outcome is always an even number. For instance, 6× 4 = 24.
• When an even number is multiplied by an odd integer, the result is always an even number. For instance, 7 ×4 = 28.
• When two odd numbers are multiplied, the outcome is always an odd number. For instance, 7 ×3 =21.

Generalization of Odd and Even Numbers:-

We can also generalize even and odd numbers. If ‘n’ is an even number, then ‘n + 1’ is the next odd number, and ‘n + 2’ is the next even number, and so on. If ‘n’ is an odd number, the following even number is ‘n + 1,’ the next odd number is ‘n + 2,’ and so on.

For instance, if we want to create a sequence of five odd numbers beginning with 33, we can do so as follows:

 33, 33 + 2, 33 + 4, 33 + 6, 33 + 8, 

 33, 35, 37, 39, 41

Numbers Chart

The number chart from 1 to 100 is shown below,  odd numbers are written in yellow and even numbers are written in green.

Even numbers :-2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100

Odd numbers:-

1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99

How to Find Even and Odd Numbers?

We can use one of the following methods to determine whether a given integer is even or odd:

• We can readily distinguish odd and even numbers by looking at the digit in the ‘ones’ location of the provided number. Odd numbers conclude with 1,3,5,7,9, while even numbers end with 0,2,4,6,8.
• Equal grouping: Even numbers can be grouped in pairs, while odd numbers cannot.

Tips on Even and Odd Numbers:-

• Each alternate number beginning with 1 is an odd number, whereas each alternate number beginning with 2 is an even number.
• Checking a number’s divisibility by two is a simple approach to determine if it is even or odd.
• Non-integer numbers, such as ½, 0.88, or infinity, are neither even nor odd.

What are Even and Odd Decimals?

Decimals aren’t whole numbers, and they can’t be even or odd. We can’t declare that the fraction 1/9 is odd because the denominator is an odd number, or that the fraction 12.38 is even because the last digit is even. Only integers can be even or odd, hence decimals and fractions cannot be even or odd. Zero, on the other hand, is an even number because it is an integer and divisible by two. Numbers that are not equally divisible by two are known as odd number.