LCM by Division Method
LCM by division method means finding the least common multiple of a given set of numbers by dividing all the given numbers by a common prime number. This is one of the easiest methods to find the LCM of numbers. Let us learn about the LCM by division method on this page.
1.  What is LCM by Division Method? 
2.  LCM by Common Division Method 
3.  HCF and LCM by Division Method 
4.  FAQs on LCM by Division Method 
What is LCM by Division Method?
LCM by division method is a method of finding the LCM (Least Common Multiple) of numbers by dividing them by common prime numbers. This method of finding the LCM of numbers is one of the most common methods which gives the result quickly. For this, we need to know the common multiplication tables and those prime numbers that can divide the given numbers completely.
LCM by Common Division Method
In order to find the LCM by the common division method, we need to know the prime factors of the given numbers. Let us see the how to calculate the LCM of 24, 36, and 48 using the division method using the steps given below:
 Step 1: Write the given numbers in one line separated by commas and draw a line under them such that we can write the quotient beneath each one of them.
 Step 2: Look for a common prime number which is a factor of the given numbers. Write this prime number on the left of the given numbers as shown in the figure given above. If the prime number which is selected is a factor of the numbers, then divide the numbers by it and write the quotient below them as shown. If the selected prime number is not a factor of any of the numbers, then copy that number in the row below it. Continue the steps until 1 is left in the last row.
 Step 3: In this case, since 2 is a prime number that divides 24, 36, and 48 we write 2 on the left side. We divide these numbers by 2 and write the quotient below each one of them.
 Step 4: In the next row, we again divide the numbers by a common prime number that divides them. In this case, we again get 2 as the divisor and we divide 12, 18, and 24 by it. We get 6, 9 and 12. We repeat the steps and look for a prime number that divides these numbers.
 Step 5: We get 2 as the divisor again and we divide 6 and 12 by it write the quotients below it. We copy the remaining number 9 to divide it in the later steps. We again divide this row of numbers by 2 to get 3, 9 and 3.
 Step 6: Now, we get the prime number 3 which can divide all these numbers. So, we write 3 on the left side and divide 3, 9, and 3 by it and write the quotients below. This brings us to 1, 3, and 1. We again divide it by 3 and we get 1, 1, and 1 which means that we can stop now.
 Step 7: Now, we multiply these prime numbers that are written on the left to get the LCM of the numbers 24, 36 and 48. This will be, 2 × 2 × 2 × 2 × 3 × 3 = 144. Therefore, the LCM of 24, 36 and 48 by division method is 144.
HCF and LCM by Division Method
In order to find the LCM by division method, we use the same steps given above. However, if we need to find the HCF by long division method, it has a different working. Let us understand this using an example for both.
Example: Find the HCF and LCM of 12 and 18 by division method.
Solution: First let us find the LCM of 12 and 18 using the steps discussed below.
 Step 1: Write the given numbers in one line separated by commas and look for a common prime number which is a factor of the given numbers. Write this prime number on the left of the given numbers as shown in the figure given above. Here, the common prime factor of 12 and 18 is 2.
 Step 2: Divide the numbers 12 and 18 by 2 and write the quotient below them as shown. We get 6 and 9 and we divide these by 2 again. Since 6 is divisible by 2 we write the quotient 3 below 6, while 9 will be copied down since it is not divisible by 2. This will give us 3 and 9 in the next row. Then, we get 3 as the prime number that divides 3 and 9. So we divide 3 and 9 by 3 to get 1 and 3. Continue the steps until 1 is left in the last row.
 Step 3: Finally, we can multiply these prime numbers that are written on the left to get the LCM of the numbers 12 and 18. This will be, 2 × 2 × 3 × 3 = 36. Therefore, the LCM of 12 and 18 by division method is 36.
Now, let us find the HCF (Highest Common Factor) of 12 and 18 by long division method using the following steps.
It should be noted that the HCF of 12 and 18 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.
 Step 1: Divide the larger number (18) by the smaller number (12).
 Step 2: Since the remainder ≠ 0, we will divide the divisor (12) of the previous step by the remainder (6).
 Step 3: Repeat this process until the remainder = 0.
 Step 4: The divisor (6) is the HCF of 12 and 18.
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LCM by Division Method Examples

Example 1: Find the LCM of 24 and 36 by division method:
Solution: The LCM by division method can be calculated using the following steps.
 Step 1: Write the numbers separated by commas as shown in the figure given above. Here, the common prime factor of 24 and 36 is 2.
 Step 2: Divide the numbers by 2 to get 12 and 18. Then, divide 12 and 18 by 2 and write the quotient below them as shown to get 6 and 9. We again divide 6 and 9 by 2. Since only 6 is divisible by 2, we write the quotient and copy 9 as it is in the next row to get 3 and 9. Then, we get 3 as the prime number that divides 3 and 9. So we divide 3 and 9 by 3 to get 1 and 3. We continue the steps until 1 is left in the last row.
 Step 3: Now, we can multiply these prime numbers that are written on the left to get the LCM of the numbers 24 and 36. This will be, 2 × 2 × 2 × 3 × 3 = 72. Therefore, the LCM of 24 and 36 by division method is 72.

Example 2: State true or false:
a.) In order to find the LCM by division method, we find a common prime number that divides the given numbers.
b.) The LCM of 12 and 15 is 60.
Solution:
a.) True, in order to find the LCM by division method, we find a common prime number that divides the given numbers.
b.) True, the LCM of 12 and 15 is 60.

Example 3: Find the LCM of the following numbers by division method: 62 and 108
Solution: The LCM of 62 and 108 can be calculated as follows.
 Step 1: Write the given numbers in one line separated by commas and find a common prime number which is a factor of the given numbers. Write this prime number on the left of the given numbers as shown in the figure given above. Here, the common prime factor of 62 and 108 is 2.
 Step 2: Divide the numbers 62 and 108 by 2 and write the quotient below them as shown. We get 31 and 54. Then, we divide this row by 2 again. We copy down 31 to the next row and get 27 below 54. We continue the steps until 1 is left in the last row.
 Step 3: Now, we can multiply these prime numbers that are written on the left and then we can get the LCM of the numbers 62 and 108. This will be, 2 × 2 × 3 × 3 × 3 × 31 = 3348. Therefore, the LCM of 62 and 108 by division method is 3348.
FAQs on LCM by Division Method
How to Find LCM by Division Method?
LCM by division method means finding the least common multiple of a given set of numbers by dividing the numbers by a common prime number. For this, we need to know the prime factors of the given numbers.
 We write the given numbers in one line separated by commas and draw a line under them such that we can write the quotient beneath each one of them.
 Then, we look for a common prime number which is a factor of the given numbers. We write this prime number on the left of the given numbers.
 If the prime number which is selected is a factor of the numbers, then we divide the numbers by it and write the quotient below them.
 If the selected prime number is not a factor of the number, then we copy the number in the row below it.
 We continue these steps until 1 is left in the last row.
 Then, we multiply these prime numbers that are written on the left to get the LCM of the numbers.
What is the LCM of 24 and 48 by Division Method?
The LCM of 24 and 48 is 48. Using the division method, we can use the following steps.
 After writing 24 and 48 in a line separating them with a comma, we look for a common prime number that divides both the numbers.
 We get 2 as the prime factor and after dividing 24 and 48 we get 12 and 24 which is written below these numbers. We continue dividing these numbers by prime numbers until we get 1 as the quotient for both the numbers. After these steps, we multiply the prime numbers written on the left and we get, 2 × 2 × 2 × 2 × 3 = 48
How to Find the LCM of 102, 119 and 153 by Division Method?
In order to find the LCM of 102, 119 and 153 by division method, we use the following steps:
 We write 102, 119 and 153 in a line separating them with commas and look for a common prime number that divides the given numbers.
 We get 2 as the prime factor and after dividing 102, 119 and 153 by division method we get 51, 119 and 153.
 Then, we divide these by 3 to get 17, 119 and 51.
 Then, we divide this row by 3 again to get 17, 119 and 17.
 After this, we divide this row by 7 to get 17, 17 and 17.
 Finally, we divide this row by 17 to get 1, 1, and 1. So we stop here because we get 1 as the quotient for all the numbers.
 After this step, we multiply the prime numbers written on the left and we get, 2 × 3 × 3 × 7 × 17 = 2142
How to Find HCF and LCM by Division Method?
While calculating the LCM of numbers by division method, we continue dividing the numbers until we get 1 as the quotient for all the numbers and then we multiply these divisors to get the LCM. When we calculate the HCF of numbers by long division method, we divide the larger number by the smaller number. Then, if the remainder is ≠ 0, we divide the divisor of step 1 by the remainder obtained. We repeat the procedure until we get the remainder as 0. The final divisor is taken to be the HCF of the two given numbers. This means that the HCF of 2 numbers is the divisor that we get when the remainder becomes 0 after the process of long division repeatedly.
How to Find the LCM of 3 Numbers by Division Method?
The LCM of 3 numbers can be calculated by division in the same way as we do it for 2 numbers. We separate them with commas and find the prime numbers that can divide them. We write the quotient below the numbers and we write the divisors on the left. We continue dividing each row until we get 1 as the quotient for all the numbers. Then, we multiply the divisors written on the left to get the LCM of the given numbers.
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