[Maths Class Notes] on Factors and Multiples Pdf for Exam

The two important concepts of mathematics are factors and multiples. These concepts are typically studied together as they are related to each other. The number that divides the other number exactly is called a factor. When you want to obtain a specific number you multiply a number with another number, this is called multiples of that number.

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Factors Definition 

The number that divides the other number exactly leaves the remainder as zero, it is called the factors of numbers. This can alternatively be explained as when the dividend is exactly divisible by the divisor leaving a zero remainder, then the divisor is the factor of the dividend. The common factor of every number is the number itself and the number 1. In the given image, we can observe that the number 20 is the multiple of numbers 4 and 5. Alternatively, the numbers 4 and 5 are the factors of the number 20.

When any natural number has only two factors (which are the number itself and the number 1), then such a number is termed as a prime number. The examples of factors representing prime numbers can be numbers 2, 5, 7, and so on.

What is Multiple?

A multiple can be defined as a product of the given number and any other natural number. To understand and observe the multiples, study the multiplication table.

Some of the basic observations for the multiples of numbers can be mentioned below:

  • Multiples of the number 2 will always be even numbers and end with 8, 6, 4, 2, and 0.

  • The multiples of the number 5 will always end with either 0 or 5.

If we represent the numbers in alphabets then where A and B are two numbers, we can conclude that:

  • If number A divides number B, then A is the factor of B

  • If number B is divisible by number A, then B is the multiple of A

Difference Between Factors and Multiples

To explain what is a factor and what is multiple, it is essential to understand the differences between both concepts. The differences between factors and multiples are tabulated below:

Factors 

Multiples 

The number that is the exact divisor of a given number is called that number’s factor.

The number that is obtained when two natural numbers or integers are multiplied, is called the multiple of the resultant number. 

Factors of numbers and finite. 

Multiples are infinite in number. 

The mathematical operation of division is used in how to find the factors of a number. 

The mathematical operation of division is used in how to find multiples of a number. 

The result obtained while finding a factor is less than or equal to the given number. 

The result obtained while finding a multiple is greater than or equal to the given number. 

How to Find Factors of a Number?

To find the factors of a number we have to find all the finite numbers that divide the given number in a way that no remainder is left after the operation of division. For instance, if we take an example of the number 28, then all the numbers that exactly divide the number 28 are 28, 14, 7, 4, 2, and 1. So all these numbers are the factors of number 28.

It should be noted however that every number has two essential factors namely the number itself and the number 1.

How to Find Multiples of a Number? 

Pick a number for which you want to find a multiple and then multiply it by continuing whole numbers. For example, let’s say we wish to find the multiples of 4.


Start by: 

4 x 1 = 4

4 x 2 = 8

4 x 3 = 12, and continue multiplying other whole numbers. 

Now, determine if the particular number, which in this case is 4, can be divided without a remainder. A multiple of any number should be able to be divided by the initial number. For example, 8 is a multiple of 4, and 4 x 2 = 8. Therefore, 8 divided by 4 will give us 2 as the answer. In this example, both 4 and 2 are considered to be factors of 8 and there is no remainder left either. 

What Does the Least Common Multiple Mean? 

When a number tends to be a multiple of 2 or more numbers, then it is referred to as a common multiple of the said numbers. 

Let’s take an example:

We know that 4 x 2 = 8. So, 8 becomes a multiple of both 4 and 2, which is why it is considered to be a common multiple of both those numbers. 

The smallest, other than zero, which happens to be the multiple of 2 or more numbers, is considered to be their Least Common Multiple or their LCM. In simpler words, the smallest number that tends to be divisible by all the given is the least common multiple of two or more numbers. 

For example, let’s take two numbers 6 and 8. 

Now, the multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, and so on. And the multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, and so on. From this, we can infer that the common multiples of 6 and 8 are 24, 48, etc… And out of these, the least common multiple of 8 and 6 is 24, therefore it is the LCM of 6 and 8. 

Methods to Find LCM of Two Numbers

Two primary methods can be used to find the least common multiple of two or more numbers namely, the LCM method or the Common Division Method. If you want to find LCM by using the prime factorisation method, then each number is to be expanded as a product of its prime factors. Then, using the highest power in the prime factorisation of each number, you find the product of all the different prime factors. And if you want to find LCM with the help of the common division method, then the numbers that are divisible by the number chosen in the previous step are to be divided after which you write down the quotients just below them. Then, the numbers that are not divisible, are carried forward. 

Properties of Factors and Multiples

To understand the concept of factors and multiples you should go through the types of factors and multiple examples. Also, certain distinctive properties make the concept clear and concise. Some of the major properties are mentioned below:

  • Every number has a common factor: the number 1.

  • Every number has a multiple: the number 0.

  • The concept of multiples and factors applies to only whole numbers.

  • Every number has a minimum of two factors namely the number 1 and the number itself. Number 1 is the smallest factor and the number itself is the largest factor.

  • Each number has one multiple that is the number itself.

  • For each number, the number of factors is finite and the number of multiples is infinite.

  • The number is termed a prime number if the number has only two factors namely the number itself and the number 1.

Applications of Factors and Multiples 

There are various applications and uses of factors and multiples. Multiples are used in various ways in the transactions carried out in our everyday lives as well as in complicated scientific and mathematical calculations that are fundamental to physics and computer science. Both the concepts of factors and multiples are very important to understand and comprehend.

They tend to set the foundation of various mathematical aspects and operations like division, measurement, patterns, etc. And if one understands the concept of factors and multiples and the relationship between them, then it would become much easier for them to navigate the overall relationship of numbers in real life. 

Conclusion

The concept of multiples and factors are the backbone of many operations in mathematics. It is very essential to be clear with these concepts so that the other operations become simple and easy.

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