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LCM of 9 and 12

LCM of 9 and 12 is the smallest number among all common multiples of 9 and 12. The first few multiples of 9 and 12 are (9, 18, 27, 36, 45, . . . ) and (12, 24, 36, 48, 60, . . . ) respectively. There are 3 commonly used methods to find LCM of 9 and 12 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 9 and 12
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 9 and 12?

Answer: LCM of 9 and 12 is 36.

Explanation:

The LCM of two non-zero integers, x(9) and y(12), is the smallest positive integer m(36) that is divisible by both x(9) and y(12) without any remainder.

Methods to Find LCM of 9 and 12

The methods to find the LCM of 9 and 12 are explained below.

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 9 and 12 by Listing Multiples

To calculate the LCM of 9 and 12 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 9 (9, 18, 27, 36, 45, . . . ) and 12 (12, 24, 36, 48, 60, . . . . )
Step 2: The common multiples from the multiples of 9 and 12 are 36, 72, . . .
Step 3: The smallest common multiple of 9 and 12 is 36.

∴ The least common multiple of 9 and 12 = 36.

LCM of 9 and 12 by Division Method

To calculate the LCM of 9 and 12 by the division method, we will divide the numbers(9, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 9 and 12.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 9 and 12. Write this prime number(2) on the left of the given numbers(9 and 12), separated as per the ladder arrangement.
Step 2: If any of the given numbers (9, 12) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 9 and 12 is the product of all prime numbers on the left, i.e. LCM(9, 12) by division method = 2 × 2 × 3 × 3 = 36.

LCM of 9 and 12 by Prime Factorization

Prime factorization of 9 and 12 is (3 × 3) = 3² and (2 × 2 × 3) = 2² × 3¹ respectively. LCM of 9 and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² = 36.
Hence, the LCM of 9 and 12 by prime factorization is 36.

☛ Also Check:

LCM of 9 and 12 Examples

Example 1: Verify the relationship between GCF and LCM of 9 and 12.

Solution:

The relation between GCF and LCM of 9 and 12 is given as,
LCM(9, 12) × GCF(9, 12) = Product of 9, 12
Prime factorization of 9 and 12 is given as, 9 = (3 × 3) = 3² and 12 = (2 × 2 × 3) = 2² × 3¹
LCM(9, 12) = 36
GCF(9, 12) = 3
LHS = LCM(9, 12) × GCF(9, 12) = 36 × 3 = 108
RHS = Product of 9, 12 = 9 × 12 = 108
⇒ LHS = RHS = 108
Hence, verified.
Example 2: The product of two numbers is 108. If their GCD is 3, what is their LCM?

Solution:

Given: GCD = 3
product of numbers = 108
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 108/3
Therefore, the LCM is 36.
The probable combination for the given case is LCM(9, 12) = 36.
Example 3: The GCD and LCM of two numbers are 3 and 36 respectively. If one number is 12, find the other number.

Solution:

Let the other number be y.
∵ GCD × LCM = 12 × y
⇒ y = (GCD × LCM)/12
⇒ y = (3 × 36)/12
⇒ y = 9
Therefore, the other number is 9.

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FAQs on LCM of 9 and 12

What is the LCM of 9 and 12?

The LCM of 9 and 12 is 36. To find the least common multiple of 9 and 12, we need to find the multiples of 9 and 12 (multiples of 9 = 9, 18, 27, 36; multiples of 12 = 12, 24, 36, 48) and choose the smallest multiple that is exactly divisible by 9 and 12, i.e., 36.

What is the Relation Between GCF and LCM of 9, 12?

The following equation can be used to express the relation between GCF and LCM of 9 and 12, i.e. GCF × LCM = 9 × 12.

Which of the following is the LCM of 9 and 12? 50, 15, 45, 36

The value of LCM of 9, 12 is the smallest common multiple of 9 and 12. The number satisfying the given condition is 36.

If the LCM of 12 and 9 is 36, Find its GCF.

LCM(12, 9) × GCF(12, 9) = 12 × 9
Since the LCM of 12 and 9 = 36
⇒ 36 × GCF(12, 9) = 108
Therefore, the GCF = 108/36 = 3.

What are the Methods to Find LCM of 9 and 12?

The commonly used methods to find the LCM of 9 and 12 are:

Prime Factorization Method
Division Method
Listing Multiples

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