LCM

How To Find LCM of 9 and 12? | Listing, Division, and Prime Factorization Method

Updated on: 17 Feb 2023

How To Find LCM of 9 and 12? | Listing, Division, and Prime Factorization Method

LCM of 9 and 12 is 36. LCM of 9 and 12, also known as Least Common Multiple or Lowest Common Multiple of 9 and 12 is the lowest possible common number that is divisible by 9 and 12.

Now, let’s see how to find the LCM of 9 and 12. Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108,… and multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108,… Here, both 36 and 72 are the common numbers in the multiples of 9 and 12, respectively, or that are divisible by 9 and 12. But, when you have to find the LCM, you must focus on the lowest common number. So, 36 is the lowest common number among all the multiples that is divisible by 9 and 12, and hence the LCM of 9 and 12 is 36.

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Methods to find the LCM of 9 and 12

There are three different methods for finding the LCM of 9 and 12. They are:

Prime Factorization Method
Listing Method
Division Method

LCM of 9 and 12 using the Prime Factorization Method

The prime factorization method is one of the methods for finding the LCM. To find the LCM of 9 and 12 using the prime factorization method, follow the following steps:

Step 1: Find the prime factors of 9 and 12 using the repeated division method.
Step 2: Write all the prime factors in their exponent forms. Then multiply the prime factors having the highest power.
Step 3: The final result after multiplication will be the LCM of 9 and 12.

LCM of 9 and 12 can be obtained using the prime factorization method as:

Prime factorization of 9 can be expressed as 3 * 3 = 3¹ * 3¹= 3²
Prime factorization of 12 can be expressed as 2 * 2 * 3 = 2¹ * 2¹ * 3¹ = 2²* 3¹
So, the LCM of 9 and 12 = 2²* 3² = 2 * 2 * 3 * 3 = 36

LCM of 9 and 12 using the Listing Method

The listing method is one of the methods for finding the LCM. To find the LCM of 9 and 12 using the listing method, follow the following steps:

Step 1: Write down the first few multiples of 9 and 12 separately.
Step 2: Out of all the multiples of 9 and 12 focus on the multiples that are common to both numbers, that is, 9 and 12.
Step 3: Now, out of all the common multiples, take out the smallest common multiple. That will be the LCM of 9 and 12.

LCM of 9 and 12 can be obtained using the listing method:

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108,…
Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108,…
Here, it is clear that the least common multiple is 36. So, the LCM of 9 and 12 is 36.

LCM of 9 and 12 Using the Division Method

The division method is one of the methods for finding the LCM. To find the LCM of 9 and 12 using the division method, divide 9 and 12 by the smallest prime number, which is divisible by any of them. Then, the prime factors further obtained will be used to calculate the final LCM of 9 and 12.

Follow the following steps to find the LCM of 9 and 12 using the division method:

Step 1: Write the numbers for which you have to find the LCM, that is 9 and 12 in this case, separated by commas.
Step 2: Now, find the smallest prime number which is divisible by either 9 or 12.
Step 3: If any of the numbers among 9 and 12 is not divisible by the respective prime number, write that number in the next row just below it and proceed further.
Step 4: Continue dividing the numbers obtained after each step by the prime numbers, until you get the result as 1 in the entire row.
Step 5: Now, multiply all the prime numbers and the final result will be the LCM of 9 and 12.

LCM of 9 and 12 can be obtained using the division method:

Prime Factors	First Number	Second Number
2	9	12
2	9	6
3	9	3
3	3	1
	1	1

So, LCM of 9 and 12 = 2 * 2 * 3 * 3 = 36

What Is the Formula for Finding the LCM of 9 and 12?

LCM of 9 and 12 can be calculated using the formula:

LCM (9, 12) = (9 * 12) / HCF (9, 12),
where HCF is the highest common factor or the greatest common divisor of 9 and 12.

Another formula, using which the LCM of 9 and 12 can be found:
9 * 12 = LCM (9, 12) * HCF (9, 12), that is,

The product of 9 and 12 is equal to the product of its LCM and HCF.

Problems Based on LCM of 9 and 12

Question 1: What is the LCM of 4, 9, and 12?
Solution:
We will find the LCM of 4, 9, and 12 using the division method:
To find the LCM of 4, 9, and 12 using the division method, first, we will find the smallest prime number which is divisible by either 4 or 9 or 12. If any of the numbers among 4, 9, and 12 is not divisible by the respective prime number, we will write that number in the next row just below it and proceed further. We will continue dividing the numbers obtained after each step by the prime numbers until we get the result as 1 in the entire row. We will multiply all the prime numbers and the final result will be the LCM of 4, 9, and 12.

Prime Factors	First Number	Second Number	Third Number
2	4	9	12
2	2	9	6
3	1	9	3
3	1	3	1
	1	1	1

So, LCM of 4, 9, and 12 = 2 * 2 * 3 * 3 = 36

Question 2: If the LCM of two numbers is 36, HCF is 3, and one of the numbers is 12, find the other number.
Solution:
As we know,
the product of two numbers = LCM * HCF
It is given that,
one of the numbers = 12, LCM = 36, and HCF = 3
Let the other number be x.
So, 12 * x = 36 * 3
x = (36 * 3) / 12
x = 108 / 12
x = 9
Hence, the other number is 9.

Question 3: Find the LCM of 9 and 12 using the prime factorization method.
Solution:
To find the LCM of 9 and 12 using the prime factorization method, first, we will find the prime factors of 9 and 12 using the repeated division method. Then, we will write all the prime factors in their exponent forms and multiply the prime factors having the highest power. The final result after multiplication will be the LCM of 9 and 12.

Prime factorization of 9 can be expressed as 3 * 3 = 3¹ * 3¹= 3²
Prime factorization of 12 can be expressed as 2 * 2 * 3 = 2¹ * 2¹ * 3¹ = 2²* 3¹
So, the LCM of 9 and 12 = 2²* 3² = 2 * 2 * 3 * 3 = 36

Question 4: Find the LCM of 9 and 12 using the division method.

Solution:

To find the LCM of 9 and 12 using the division method, first, we will find the smallest prime number which is divisible by either 9 or 12. If any of the numbers among 9 and 12 is not divisible by the respective prime number, we will write that number in the next row just below it and proceed further. We will continue dividing the numbers obtained after each step by the prime numbers until we get the result as 1 in the entire row. We will multiply all the prime numbers and the final result will be the LCM of 9 and 12.

Prime Factors	First Number	Second Number
2	9	12
2	9	6
3	9	3
3	3	1
	1	1

So, LCM of 9 and 12 = 2 * 2 * 3 * 3 = 36

Question 5: What is the least possible number which is divisible by 9 and 12?
Solution:
There are three methods using which you can find the least possible number which is divisible by 9 and 12.
We will find it using the listing method. First, we will write down the first few multiples of 9 and 12 separately. Out of all the multiples of 9 and 12, we will focus on the multiples which are common to both numbers.
Then, out of all the common multiples, we will take out the smallest common multiple. That will be the LCM of 9 and 12.

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108,…
Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108,…
Here, it is clear that the least common multiple is 36.

So, the LCM of 9 and 12 is 36.

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Frequently Asked Questions (FAQs)

Is 72 also considered as the LCM of 9 and 12?
72 isn’t considered as the LCM of 9 and 12. 72 is a common multiple of 9 and 12. But, it is not the least common number which is divisible by 9 and 12, and while finding the LCM, you must focus on the lowest common number. So, 36 is the lowest common number divisible by 9 and 12.

Are LCM and HCF of 9 and 12 the same?
LCM of 9 and 12 is 36 and HCF of 9 and 12 is 3. So, LCM and HCF of 9 and 12 are not the same.

What is the LCM of 9 and 12?
LCM of 9 and 12 is 36.

What are the methods to find the LCM of 9 and 12?
There are 3 major methods for finding the LCM of 9 and 12:

Prime Factorization Method
Listing Method
Division Method

Are the LCM of 9 and 12 the same as the LCM of 4, 9, and 12?
LCM of 9 and 12 is 36 and LCM of 4, 9, and 12 is also 36. So, the LCM of 9 and 12 are the same as the LCM of 4, 9, and 12.

We hope you understand all the basics on how to find the LCM of 9 and 12.

Written by

Prerit Jain

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