Multiples

The multiplication table of a number is a list of the multiples of a given number.

 In mathematics, multiples of a number are the values we get when we multiply natural numbers with the given number.

Let's take an example 6 × 5 = 30, where 30 is a multiple of both 6 and 5.

To trace the history of multiples, we need to go back thousands of years to the ancient times of the Babylonians.

In ancient civilizations like the Egyptians, Babylonians, and Greeks, multiples were utilized in numerous ways and fields, like trade, construction, and calculating time. 

To consider a number as a multiple of any number, it should have certain properties. There are facts you might not know about the characteristics of a multiple.

 
We will now point out these properties to help you understand the concept better:

• The multiples of a number can be greater or equal to the given number. Example: 5 × 1= 5, 5 × 2= 10, 5 × 3= 15. 

• Every number has an infinite number of multiples, which cannot be fully listed. For example, if we need to list the multiples of 4, we start with 4, 8, 12, 16, and so on. Since the multiples of a number are endless, we cannot include all of them here.

• The multiples of a number always begin with the number itself. For example, when you check the multiples of any given number, you find that 2 ×1= 2, 3×1= 3. This property applies to all numbers.

• The multiples of any number follow a certain pattern or flow. For example, the multiples of 5 always end with either 5 or 0 i.e., 5, 10, 15, 20, and so on.
   

As we have learned the definition, history, and properties of multiples, we now understand multiples have a specific pattern. Multiples of a number can vary depending on the type. To recognize the pattern of multiples or to solve problems related to multiples, we need to learn its different types.

The different types of multiples are mentioned below:

Even multiples

If the multiples of a given number are the same as the multiples of 2 and end in either 2, 4, 6, or 8, then they are even multiples.

For example, 2, 4, 6, 8, etc. are even multiples. They follow a pattern where the difference between each multiple is 2.

Odd multiples

Odd multiples refer to the list of multiples that are not divisible by 2.

For example, 3, 9, 15, 21, etc., are odd multiples.

Common multiples

As the name suggests, common multiples can be the multiples of 2 or more numbers. For example: Let’s take multiples of 4 and 6:
Multiples of 4 are 4, 8, 12, 16,...
Multiples of 6 are 6, 12, 18, 24,…
The common multiples of the numbers 4 and 6: 12, 24, 36,…
 

Compared to ancient times, multiplication is now learned and understood by many. Mathematics is all around us, and so are multiples. Understanding the concept of multiples of a number helps students overcome real-life challenges, such as calculating budgets, and dealing with change. 

We will now list down a few key benefits of understanding multiples:

• Learning multiples helps students enhance their logical thinking skills, as it allows them to perform mind math.

• Students can apply their knowledge of multiples in real-life situations, such as calculating costs or distributing equal shares. 

• Having a solid understanding of the concept helps them solve the problem effortlessly, thereby improving their confidence.
   

We have now learned the importance of understanding multiples in general. Next, we will discuss how students can apply their knowledge of multiples in mathematics. Learning multiples lays the foundation for other complex concepts in math. 

Here are a few applications of multiples in math:

Factors and Multiples

Factors and multiples are two related but different concepts. Factors are the numbers that divide a given number exactly without any remainder. Whereas, multiples of a number are the products we get from multiplying the given number with a set of natural numbers. Understanding the multiples helps students recognize the patterns in numbers.

For example:
Take the number 8
The factors of 8 are 1, 2, 4, and 8
The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, and so on, …

Highest Common Factor (HCF) Using Multiples

The HCF of two or more numbers is the largest common factor dividing the given numbers completely. The common factors are found using the multiples of given numbers.

For example: to find the HCF of 15 and 25 we do the following steps:
Multiples of 15 are 15, 30, 45, 60, etc.
Multiples of 25 are 25, 50, 75, 100,...
The common factors of 15 and 25 are 1 and 5
Therefore, we take 5 as the HCF

The Least Common Multiple (LCM) Using Multiples

The LCM or the least common multiple, is the smallest number by which two or more numbers can be completely divided.

For example: We find the LCM of 6 and 8
Multiples of 6 are 6, 12, 18, 24,...
Multiples of 8 are 8, 16, 24, 32,...
Considering the common factors, we get 24 as the LCM of 6 and 8.

Prime Factorization and Multiples

Prime factorization of a number is the process of dividing a given number into its prime factors. This helps us in determining its multiples. We obtain multiples in the process of multiplying the prime factors with natural numbers.

For example: The prime factorization of 46 is 2 × 23. Here, 2 and 23 are the prime factors of 46.

Multiples of 46 are 46, 92, 138, 184,...
 

As we have learned the importance of multiple, we know how important it is to learn multiples. To master the concept of multiples, students can use certain tips and tricks. Let’s look into some:

Learn Times Table

The first and most important tip is to learn multiplication tables up to 12 at least. It can also help students learn the multiples of larger numbers. Learning multiples can even help you perform mental math in no time.

Incorporate Real-life Examples

Multiples can be utilized in several real-life situations. When we have to deal with money. For example, when we have to buy 10 chocolates that cost 5 rupees each, you need to use multiplication to find the total cost of the chocolates, which is 50 rupees.

Learn Through Card Games

Students learn better when the multiples of a number are displayed on flashcards. They can practice the multiples by reading from it. They can also learn multiples to number games such as bingo cards, where the multiples are written. Learning through such games is a quick way to recall the multiples of any number.

Recognize Patterns

Understanding patterns of multiples is a simple but effective way to master multiples. It is important to learn what patterns are. 
For example: 
The multiples of 2 always ends in the numbers 0, 2, 4, 6, or 8 
The multiples of 10 always end with 0. 

Make it smaller

Breaking the larger number into smaller units makes it easier to multiply. 
For example:

12 × 15 can be written as (12 × 10) + (12 × 5)
That is 120 + 60, which is equal to 180