# Learning Objectives of the chapter Factors and Multiples for Class 4

**At the end of this chapter you will be able to:**

## Multiples Of A Number

Below I am giving some examples of multiples of numbers from 2 to 9.

- Multiple of 2:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20 - Multiple of 3:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30 - Multiple of 4:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40 - Multiple of 5:

5, 10, 15, 20, 25, 30, 35, 40, 45, 50 - Multiple of 6:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60 - Multiple of 7:

7, 14, 21, 28, 35, 42, 49, 56, 63, 70 - Multiple of 8:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80 - Multiple of 9:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90

**Also Read:**
Roman Numerals chapter for Class 4

**Now, let see the following example.**

12 = 1 x 12 or 12 = 12 x 1

12 = 2 x 6 or 12 = 6 x 2

12 = 3 x 4 or 12 = 4 x 3

Here, we can say that 12 is a **multiple **of 1, 2, 3, 4, 6 and 12.

**Let's take an another example.**

18 = 1 x 18 or 18 = 18 x 1

18 = 2 x 9 or 18 = 9 x 2

18 = 3 x 6 or 18 = 6 x 3

Thus, 18 is a multiple of 1, 2, 3 and 6

**Properties of Multiples**

(1) 6 x 1 = 6, 9 x 1 = 9, 15 x 1 = 15

In the above example, you can see that,

6 is a multiple of 6 and 1, 9 is a multiple of 9 and 1, 15 is a multiple of 15 and 1.

**Thus, every number is a multiple of itself and 1.**

(2) 18 is a multiple of 1, 2, 3, 6 and 18.

See carefully that 18 is greater than 1, 2, 3 and 6. And it is equal 18.

**Thus, multiple of a number is always greater than or equal to the number itself.**

(3) Let's take a number 7. To find its multiple, we can multiply 7 by higher and higher numbers to get more and more multiples. Thus, we can get uncountable number of multiples.

**Thus, a number has uncountable number of multiples and there is no largest multiple of a number.**

**From the above example you can see that,**

- Every number is a multiple of number 1.
- Every number is a multiple of itself.
- A multiple of a number is always greater than or equal to the number itself.
- To find multiple of a number we can multiply it by 1, 2, 3, 4, 5, 6, ...
- A number has uncountable number of multiples.
- There is no largest multiple of number as we can multiply a number by any number.

**Exercise on Multiples**

20 = 1 x 20 or 20 x 1

20 = 2 x 10 or 10 x 2

20 = 4 x 5 or 5 x 4

Thus, 20 is a multiple of 1, 2, 4, 5, 10 and 20.

So, the answer would be 2, 4, 5.

5 x 8 = 40

Thus, 40 is a multiple of both 5 and 8.

True.

For example,

21 is a multiple of 1, 3, 7 and 21.

Thus, 21 is a multiple of itself also.

False.

Every number is a multiple of 1.

## Factors Of A Number

A **Factor **of a number divides the number without leaving a remainder.

Let's see an example below.

18 is a **multiple **of 1, 2, 3, 6 and 18.

If we divide 18 by 1, 2, 3, 6 or 18, no remainder is left. That means, we get 0 as remainder.

Thus, we can say that 1, 2, 3, 6 and 18 are **factors **of 18.

**We can also find factor by using multiplication.**

Let's take another example.

2 x 9 = 18, here, 2 and 9 are factors of 18.

4 x 7 = 28, here, 4 and 7 are factors of 28.

5 x 6 = 30, here, 5 and 6 are factors of 30.

**REMEMBER**

- When two or more numbers are multiplied we get a
**product**. - The product is a
**multiple**of each of the numbers multiplied. - Each number is a
**factor**of the product.

For example, 3 x 8 = 24, here, 24 is the product.

Or we can say, 24 is the multiple of both 3 and 8.

Both 3 and 8 are factors of 24.

**Properties of Factors**

(1) 1 can divide any number. Yes, any number.

**Thus, 1 is a factor of every number.****Also, 1 is the smallest factor of every number.**

(2) 9 is a multiple of 1, 3 and 9.

See carefully that 1, 3 and **9** are **factors **of **9**.

**So, every number is a factor of itself.**

(3) **A number does not have uncountable number of factors.** In other words, it has limited number of factors.

For example,

12 has 5 factors which are 1, 2, 3, 4 and 12.

15 has 4 factors which are 1, 3, 5 and 15.

(4) **A factor of a number is either equal to or smaller than the number. The biggest factor of a number is the number itself**.

Let's take the same example as above.

The factors of 15 are 1, 3, 5 and 15.

Thus, all the factors 1, 3, 5 and 15 are either equal to or smaller than 15. Also, 15 is the biggest factor of 15.

**Exercise on Factors**

20 = 1 x 20 or 20 x 1

20 = 2 x 10 or 10 x 2

20 = 4 x 5 or 5 x 4

Thus, 20 is a multiple of 1, 2, 4, 5, 10 and 20.

So, the answer would be 2, 4, 5.

5 x 8 = 40

Thus, 40 is a multiple of both 5 and 8.

True.

For example,

21 is a multiple of 1, 3, 7 and 21.

Thus, 21 is a multiple of itself also.

False.

Every number is a multiple of 1.

## Even Numbers

A number which is a multiple of 2 is called an **even number**.

For example, 2, 4, 6, 8, 10, 12, 14, ...

## Odd Numbers

A number which is not a multiple of 2 is called an **odd number**.

For example, 1, 3, 5, 7, 9, 11, 13, ...

## Prime Numbers

Numbers which have only two factors, 1 and the number itself, are called **Prime numbers**.

For example, 2, 3, 5, 7, 11, 13, 17, 19 are examples of prime numbers.

## Composite Numbers

Numbers which have more than two factors are called **Composite numbers**.

For example, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20 are examples of composite numbers.

**Exercise on Even, Odd, Prime and Composite Numbers**

False.

Each prime numbers has exactly two factors, 1 and the number itself.

1.

The number 1 is neither prime nor composite.

97

13, 23 and 31 are prime numbers as they have only two factors, 1 and the number itself.

## Common Factors

We already know how to find factors of a number. Now let's learn how to find common factors of numbers with the following example.

**Exampe 1: Find the common factors of 15 and 21.**

Factors of 15 are **1**, **3**, 5 and 15

Factors of 21 are **1**, **3**, 7 and 21

Common factors of 15 and 21 are **1 and 3**.

**Exampe 2: Find the common factors of 27 and 45.**

Factors of 27 are **1**, **3**, **9** and 27

Factors of 45 are **1**, **3**, 5, **9**, 15 and 45

Common Factors of 27 and 45 are **1, 3 and 9**.

## Highest Common Factors (HCF)

**Example 1:** Let's take the example 1 above and find the highest common factor of 15 and 21.

The common factors of 15 and 21 are 1 and 3.

**Here, the highest common factors of 15 and 21 is 3.**

**Example 2:** Find HCF of 27 and 45.

Common factors of 27 and 45 are 1, 3 and 9.

Thus, HCF of 27 and 45 is 9.

**Example 3:** Find HCF of 12, 16 and 24.

Factors of 12 are 1, 2, 3, 4, 6 and 12

Factors of 16 are 1, 2, 4, 8 and 16

Factors of 24 are 1, 2, 3, 4, 6, 12 and 24

Common factors of 12, 16 and 24 are 1, 2 and 4.

Thus, HCF of 12, 16 and 24 is 4.

**Example 4:** Find HCF of 4 and 16.

In this example, please note that **4 is a factor of 16**.

Factors of 4 are 1 and 4

Factors of 16 are 1, 4 , 8 and 16

Common factors of 4 and 16 are 1 and 4.

Thus, HCF of 4 and 16 is 4.

If a number is a factor of another number, then their HCF is the smaller of the two numbers.

In above example, 4 is a factor of 16. Also, out of 4 and 16, 4 is the smaller number, therefore, 4 is the HCF of 4 and 16.

**Example 5:** Find the HCF of 8 and 15.

Factors of 8 are 1, 2, 4 and 8

Factors of 15 are 1, 3, 5 and 15

Common factors of 8 and 15 is only 1.

Thus, there is no highest common factor. Also, two numbers which have only 1 as the common factor are called **coprime**.

Therefore, 8 and 15 are coprime.

Nice