**Introduction**

Multiplication and division are two of the important operations in mathematics. We can multiply a number by any number. Similarly, a number can be divided by any number. Two important terms that are related to the multiplication and division of numbers are factors and multiples. The understanding of one of these is incomplete without studying the other. Therefore, before we move ahead and learn about multiples let us know what we mean by factors of a number.

**What are Factors?**

**A factor of a number is an exact divisor of that number. **In other words, a factor of a number is that number that completely divides the number without leaving a remainder. For example, each of the numbers, 1, 2, 3, 4, 6 and 12 is a factor of 12. However, none of the numbers 5, 7, 8, 9, 10 and 11 is a factor of 12.

**What is a multiple?**

A multiple of a whole number is the product of the number and any counting number. If we multiply 3 by 1, 2 , 3 , 4, 5 , 6 ….. we get

3 x 1 = 3

3 x 2 = 6

3 x 3 = 9

3 x 4 = 12

3 x 5 = 15

3 x 6 = 18 etc.

Thus 3, 6 , 9 , 12 , 15 , 18 and so on are the multiples of 3.

Let us now look at the properties that are satisfied by the multiples.

**Properties of Multiples**

- Every multiple of a number is greater than or equal to that number. For example, the multiples of 5 are 5, 10, 15, 20, 25, ……. . In fact, the multiples of a number are obtained by multiplying the number by 1, 2, 3, 4, 5, 6 …. and so on. Therefore, the smallest multiple of a number is the number itself. Hence, every multiple of a number is greater than or equal to the number itself.
- The smallest multiple of a number is the number itself. Since every number can be multiplied by 1 to get the same number, therefore, every number is a multiple of itself.
- The number of multiples of a given number is infinite. For example, the multiples of 7 are 7, 14, 21, 28, 35 and so on. We can see that this is a never ending list. Hence, the number of multiples of a given number is infinite.
- All multiples of 2 are even numbers. We know that 2, 4, 6, 8, 10, 12 and so on are the multiples of 2. Also, all these numbers 2, 4, 6, 8, 10, 12 ….. are even numbers. Hence, all multiples of 2 are even numbers.

Let us understand multiples through an example.

**Example**

Write first five multiples of 17.

**Solution**

In order to obtain tehf irst five numbers of 17, we will multiply it by 1, 2, 3, and 5. We will get –

17 x 1 = 17

17 x 2 = 34

17 x 3 = 51

17 x 4 = 68

17 x 5 = 85

**Hence, the first five multiples of 17 are 17, 34, 51, 68 and 85.**

**Common Multiples of Numbers**

We have now understood what we mean by multiples of numbers. Now, if we look at two or more numbers simultaneously, we can obtain some common multiples of those numbers. For instance, we if have two numbers 4 and 6, we can easily say that 12 is a common multiple of both 4 and 6 as 4 x 3 = 12 and 6 x2 = 12. This means that two or more numbers can have common multiples. Those multiples which are common among the multiples of two or more numbers are known as common multiples of those numbers.

Let us understand this using an example –

Example

List 3 common multiples of 3, 4 and 9

Solution

We have been given three numbers 3, 4 and 9 and we need to find common multiples of these three numbers.

Let us first write the multiples of the given numbers separately. We will have,

Multiples of 3 are 3, 6, 9, 12, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111 ……….

Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112 ……..

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117 …….

From the above multiples of 3, 4 and 9 we can see that the common multiples of 3, 4 and 6 are 36, 72 and 108.

**Hence, common multiples of 3, 4 and 6 are 36, 72 and 108.**

From above we can see that though there is no end to the list of multiples, there is always the first multiple of a number or group of numbers. Therefore, we cannot define the largest multiple of a number or a group of numbers but we can always list out the first multiple of a number. In the case of a group of numbers, this first multiple is also the least common multiple of these numbers.

Let us find out what we mean by the least common multiple of numbers.

**Least Common Multiple ( L C M )**

If a number is a multiple of two or more numbers, it is called a common multiple of the numbers. For example, we know that

2 x 3 = 6

Therefore, 6 is a multiple of both 2 and 3. Therefore 6 is said to be a common multiple of both 2 and 3.

The Least Common Multiple ( L C M ) of two or more numbers is defined as the smallest number ( other than zero ) that is a multiple of those numbers. In other words, the least common multiple of two or more numbers is the smallest number which is divisible by all the given numbers. This means that there cannot be a number divisible by the given numbers and smaller than the least common multiple.

Let us understand this through an example.

Suppose we have two numbers, 8 and 12.

Let us check the multiples of these two numbers. We have,

Multiples of 8 = 8, 16, 24, 32, 40 , 48, 56, 64, 72 and so on…..

Multiples of 12 = 12, 24, 48, 60, 72 and so on …………

From above we can see that the common multipleS of 8 and 12 are 24, 48, 72 and so on. Among these 24 is the least common multiple of these two numbers. Therefore, 12 is the least common multiple or L C M of 8 and 12.

We have just seen that by writing the multiples of each number, we can compare their multiples and find out the least common multiple among them. But is this the only way to find the least common multiple or do we have some defined steps using which we can find the L C M of two or more numbers? Let us find out.

**How to Find a Least Common Multiple ( L C M ) ?**

There are two major methods using which we can find the L C M of two or more numbers. These methods are –

- Prime Factorisation Method
- Common Division Method

Let us discuss these methods one by one.

**Prime Factorisation Method**

As the name suggests, the prime factorisation method involves finding the prime factors of the given numbers and then computing the least common multiple ( L C M ). In this method, the following steps are used to find the L C M –

- Obtain the given numbers.
- Find the prime factors of each number.
- Expand each number as a product of its prime factors
- Find the product of all different prime factors with the highest power in the prime factorisation of each number.
- The number obtained in the above step is the required L C M of the given numbers.

Let us understand the above steps using an example.

**Example**

Find the L C M of 40, 36 and 126 using the prime factorisation method.

Solution

We have been given the numbers 40, 36 and 126 and we are required to find the L C M of these numbers. Going by the above steps let us first find the prime factors of each of the given numbers. We have,

Prime factors of 40 are –

**40 = 2 x 2 x 2 x 5**

Prime factors of 36 are –

**36 = 2 x 2 x 3 x 3**

Prime factors of 126 are –

**126 = 2 x 3 x 3 x 7**

We can see in the above prime factorisations of the above numbers, the number 2 appears a maximum of three times, which is the case in the number 40. Similarly, the number 3 appears as a factor for the maximum number of 2 times, which is the case in 36 and 126. The prime factors 5 and 7 occur only in 40 and 126 respectively. Therefore, the required L C M of the numbers, 40, 36 and 126 will be

L C M of 40, 36 and 126 = 2 x 2 x 2 x 3 x 3 x 5 x 7 = 2520

**Hence, L C M of 40, 36 and 126 = 2520**

**Common Division Method**

The following steps are followed to find the L C M of two or more numbers using the common division method –

- Obtain the given numbers.
- Arrange the given numbers in a row separated by commas.
- Obtain a number that divides exactly at least two of the given numbers.
- Divide the numbers which are divisible by the number chosen in the above step and write the quotients just below them. Carry forward the numbers which are not divisible.
- Repeat the above steps till no two of the given numbers are divisible by the same number.
- Find the product of the divisors and the undivided numbers to get the required L C M of the given numbers.

Let us understand the above steps using an example.

**Example**

Find the L C M of 624 and 936 using the common division method.

**Solution**

We have been given the numbers 624 and 936 and we are required to find the L C M using 624 and 936.

We have,

From the above common division we have,

L C M of the numbers 624 and 936 = 2 x 2 x 2 x 3 x 13 x 2 x 3 = 1872

**Hence, L C M of the numbers 624 and 936 = 1872**

**Applications of Least Common Multiple ( L C M)**

We shall now have a look at some applications of L C M in solving some practical problems. Let us consider some examples.

**Example 1**** **Determine the two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 7.

**Solution**** **The smallest number which is exactly divisible by 2, 3, 4, 5, 6 and 7 is their L C M. but, we have to find two numbers nearest to 10000 which are exactly divisible by the given numbers i.e. 2, 3, 4, 5, 6 and 7. We can see that such numbers are multiples of the L C M of the given numbers. To find the L C M of 2, 3, 4, 5, 6 and 7 we have,

Therefore, the L C M of 2, 3, 4, 5, 6 and 7 is 420.

The number nearest to 10000 and exactly divisible by each of 2, 3, 4, 5, 6 and 7 should also be exactly divisible by their L C M i.e. 420. Let us now divide 10000 by 420. We find that the remainder is 340.

Number just less than 10000 and exactly divisible by 420

= 10000 – 340 = 9660.

Number just greater than 10000 and exactly divisible by 420

= 10000 + ( 420 – 340 ) = 10080

**Hence the two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 79660 and 10080.**

**Example 2** Sam has a camera that allows 24 exposures whereas Peter has a camera that allows 36 exposures. Both of them want to be able to take the same number of photographs and complete their rolls of the film. How many rolls should each buy?

**Solution** We have been given that Sam has a camera that allows 24 exposures whereas Peter has a camera that allows 36 exposures. Both of them want to be able to take the same number of photographs and complete their rolls of the film.

Since the film in Sam’s camera can take 24 exposures and the film in Peter’s camera can take 36 exposures and both of them want to take the same number of photographs while completing the rolls of the film as well, therefore, the total number of exposures taken by each will be the L C M of 36 and 24. Hence, let us find the L C M of 36 and 24.

We have,

From above, we can see that the L C M of 36 and 24 = 2 x 2 x 3 x 3 x 2 = 72

Hence, the number of rolls Sam should buy will be $\frac{72}{24}$ = 3

Similarly, the number of rolls Peter should buy will be $\frac{72}{36}$ = 2

**Key Facts and Summary**

- A factor of a number is an exact divisor of that number.
- A multiple of a whole number is the product of the number and any counting number.
- If a number is a multiple of two or more numbers, it is called a common multiple of the numbers.
- Every multiple of a number is greater than or equal to that number.
- The smallest multiple of a number is the number itself.
- The number of multiples of a given number is infinite.
- All multiples of 2 are even numbers.
- Those multiples which are common among the multiples of two or more numbers are known as common multiples of those numbers.
- We cannot define the largest multiple of a number or a group of numbers but we can always list out the first multiple of a number.
- The Least Common Multiple ( L C M ) of two or more numbers is defined as the smallest number ( other than zero ) that is a multiple of those numbers.
- There are two major methods using which we can find the L C M of two or more numbers – Least Common Method ( L C M ) and Common Division Method.
- For finding the L C M using the Prime factorization method, we expand each number as a product of its prime factors. Then we find the product of all different prime factors with the highest power in the prime factorisation of each number.
- For finding the L C M using the common division method we divide the numbers which are divisible by the number chosen in the above step and write the quotients just below them. Then we carry forward the numbers which are not divisible.

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