Odd & Even Numbers (Meaning, Properties & Examples)

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Natural numbers also known as counting numbers consist of many categories of numbers such as prime and composite numbers, even and odd numbers, etc. The classification of even and odd numbers is based on the divisibility of a number by $2$. The numbers that are completely divisible by $2$ and leave no remainder are called even numbers. On the other hand, the numbers that are not completely divisible by $2$ and leave a remainder of $1$ are called odd numbers.

Let’s learn what are even numbers.

What Are Odd and Even Numbers?

An even number is a number that is a multiple of $2$ or a number that is completely divisible by $2$. Let’s try to understand what is the meaning of a number being divisible by $2$. Suppose you have a certain number of marbles and you want to share them with your friend. You start distributing the marbles by picking a marble from your collection and keeping one with you and next giving it to your friend. You repeat the process until all marbles are finished or only $1$ marble is left that obviously, you cannot share between two people.  

If no marble is left in the collection, then it means you had an even number of marbles and if a $1$ marble is left after sharing an equal number of marbles, then it means that you had an odd number of marbles.

Odd and even numbers can also be characterized as:

• Odd Number: A number that is not divisible by $2$ and leaves $1$ as the remainder is called an odd number.
• Even Number: A number that is divisible by $2$, leaving $0$ as the remainder is called an even number.

How to Check Odd and Even Numbers?

You can check whether a given number is an even or an odd number using either of the following ways:

• By checking the digit at the ones place of the given number, you can identify an odd or an even number easily. Even numbers end with digits $0$, $2$, $4$ ,$6$, or $8$ and odd numbers end with digits $1$, $3$, $5$ , $7$, or $9$.
• By equal grouping: Even numbers can be grouped into pairs whereas odd numbers cannot be grouped in pairs.
• Divisibility by $2$: Even numbers when divided by $2$ leave $0$ as a remainder, whereas odd numbers when divided by $2$ leave $1$ as a remainder. 

Representing Odd and Even Numbers in Set Form

The set of odd and even numbers can be represented in the set-builder form. The set-builder representation of the set of odd and even numbers is:

• The set of even numbers is represented as $\{x: x = 2n \}$, where $n$ is a whole number.
• The set of odd numbers is represented as $\{x: x = 2n + 1 \}$, where $n$ is whole number.

Difference Between Odd And Even Numbers

The major differences between odd and even numbers are as follows

Parameter	Odd Numbers	Even Numbers
Representation	Numbers that can be expressed in the form of $2n + 1$, where n is a whole number	Numbers that can be expressed in the form of $2n$, where n is a whole number
Division in Pairs	Numbers that cannot be divided into two equal parts	Numbers that can be divided into two equal parts
Division by $2$	When we divide an odd number by $2$, we get the remainder $1$	When we divide an even number by $2$, we get the remainder $0$
One’s Place	All numbers ending with $1$, $3$, $5$, $7$, and $9$ are referred to as odd numbers	All numbers ending with $0$, $2$, $4$, $6$, and $8$ are referred to as even numbers
Product	The product of two odd numbers is always odd	The product of two even numbers is always even
Examples	Examples of odd numbers are $1$, $3$, $5$, and $7$	Examples of even numbers are $2$, $4$, $6$, and $8$

Can Negative Numbers Be Odd Or Even?

Negative numbers can also be classified as odd or even numbers. Here also, the same definitions of odd and even numbers hold true.

Any negative number that is divisible by $2$ and leaves a remainder of $0$ is an even number. Examples of negative even numbers are $-2$, $-4$, $-6$, and $-8$.

Any negative number that is not divisible by $2$ and leaves a remainder of $1$ is an odd number. Examples of negative even numbers are $-1$, $-3$, $-5$, and $-7$.

Can Decimals Be Odd Or Even?

The decimal numbers such as $2.8$ and $5.7$ are not either odd or even numbers. The reason is these numbers when divided by $2$ do not leave either $0$ or $1$ as remainders.

Similarly, the fractions are also not classified as odd or even numbers. The numbers such as $\frac {2}{7}$ or $\frac {3}{14}$ are neither odd nor even numbers. These numbers also don’t leave the remainder of $0$ or $1$ when divided by $2$.

Properties of Odd and Even Numbers

These are the basic properties of odd and even numbers.

• The sum of even numbers is always an even number, i.e, Even Number + Even Number = Even Number. For example, $14 + 4 = 18$.
• The sum of odd numbers is always an even number, i.e, Odd Number + Odd Number = Even Number. For example, $11 + 5 = 16$.
• The sum of an odd number and an even number is always an odd number, i.e., Even number + Odd Number = Odd Number. For example, $18 + 7 = 25$.
• The difference between even numbers is always an even number, i.e, Even Number – Even Number = Even Number. For example, $52 – 38 = 14$.
• The difference between odd numbers is always an even number, i.e, Odd Number – Odd Number = Even Number. For example, $27 – 9 = 18$.
• The difference between an odd number and an even number is always an odd number, i.e, Even Number – Odd Number = Odd Number or Odd Number – Even Number = Odd Number. For example, $35 – 24 = 11$ and $46 – 21 = 25$.
• The product of even numbers is always an even number, i.e., Even Number $\times$ Even Number = Even Number. For example, $12 \times 8 = 96$.
• The product of odd numbers is always an odd number, i.e., Odd Number $\times$ Odd Number = Odd Number. For example, $19 \times 9 = 171$.

Conclusion

Observing the remainder obtained when a number is divided by $2$ helps in determining whether a number is an odd number or an even number. The odd numbers leave a remainder of $1$ when divided by $2$, whereas the even numbers leave no remainder when divided by $2$.

Practice Problems

• Even numbers leave a remainder of $0$ when divided by $2$.
• Odd numbers leave a remainder of $0$ when divided by $2$.
• Even numbers leave a remainder of $1$ when divided by $2$.
• Odd numbers leave a remainder of $1$ when divided by $2$.
• Even numbers are divisible by $2$.
• Odd numbers are divisible by $2$.
• Even numbers are represented in the form $2n$.
• Odd numbers are represented in the form $2n$.
• Even numbers are represented in the form $2n + 1$.
• Odd numbers are represented in the form $2n + 1$.
• The sum of two even numbers is always an even number.
• The sum of two even numbers is always an odd number.
• The sum of two even numbers can be an odd number or an even number.
• The sum of two odd numbers is always an even number.
• The sum of two odd numbers is always an odd number.
• The sum of two odd numbers can be an odd number or an even number.
• The sum of an even number and an odd number is always an even number.
• The sum of an even number and an odd number is always an odd number.
• The sum of an even number and an odd number can be an even number or an odd number.

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties

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