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LCM of 54 and 12

LCM of 54 and 12 is the smallest number among all common multiples of 54 and 12. The first few multiples of 54 and 12 are (54, 108, 162, 216, . . . ) and (12, 24, 36, 48, . . . ) respectively. There are 3 commonly used methods to find LCM of 54 and 12 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 54 and 12
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 54 and 12?

Answer: LCM of 54 and 12 is 108.

Explanation:

The LCM of two non-zero integers, x(54) and y(12), is the smallest positive integer m(108) that is divisible by both x(54) and y(12) without any remainder.

Methods to Find LCM of 54 and 12

Let's look at the different methods for finding the LCM of 54 and 12.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 54 and 12 by Division Method

To calculate the LCM of 54 and 12 by the division method, we will divide the numbers(54, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 54 and 12.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 54 and 12. Write this prime number(2) on the left of the given numbers(54 and 12), separated as per the ladder arrangement.
Step 2: If any of the given numbers (54, 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 54 and 12 is the product of all prime numbers on the left, i.e. LCM(54, 12) by division method = 2 × 2 × 3 × 3 × 3 = 108.

LCM of 54 and 12 by Prime Factorization

Prime factorization of 54 and 12 is (2 × 3 × 3 × 3) = 2¹ × 3³ and (2 × 2 × 3) = 2² × 3¹ respectively. LCM of 54 and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3³ = 108.
Hence, the LCM of 54 and 12 by prime factorization is 108.

LCM of 54 and 12 by Listing Multiples

To calculate the LCM of 54 and 12 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 54 (54, 108, 162, 216, . . . ) and 12 (12, 24, 36, 48, . . . . )
Step 2: The common multiples from the multiples of 54 and 12 are 108, 216, . . .
Step 3: The smallest common multiple of 54 and 12 is 108.

∴ The least common multiple of 54 and 12 = 108.

☛ Also Check:

LCM of 54 and 12 Examples

Example 1: The GCD and LCM of two numbers are 6 and 108 respectively. If one number is 54, find the other number.

Solution:

Let the other number be b.
∵ GCD × LCM = 54 × b
⇒ b = (GCD × LCM)/54
⇒ b = (6 × 108)/54
⇒ b = 12
Therefore, the other number is 12.
Example 2: Verify the relationship between GCF and LCM of 54 and 12.

Solution:

The relation between GCF and LCM of 54 and 12 is given as,
LCM(54, 12) × GCF(54, 12) = Product of 54, 12
Prime factorization of 54 and 12 is given as, 54 = (2 × 3 × 3 × 3) = 2¹ × 3³ and 12 = (2 × 2 × 3) = 2² × 3¹
LCM(54, 12) = 108
GCF(54, 12) = 6
LHS = LCM(54, 12) × GCF(54, 12) = 108 × 6 = 648
RHS = Product of 54, 12 = 54 × 12 = 648
⇒ LHS = RHS = 648
Hence, verified.
Example 3: The product of two numbers is 648. If their GCD is 6, what is their LCM?

Solution:

Given: GCD = 6
product of numbers = 648
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 648/6
Therefore, the LCM is 108.
The probable combination for the given case is LCM(54, 12) = 108.

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FAQs on LCM of 54 and 12

What is the LCM of 54 and 12?

The LCM of 54 and 12 is 108. To find the LCM (least common multiple) of 54 and 12, we need to find the multiples of 54 and 12 (multiples of 54 = 54, 108, 162, 216; multiples of 12 = 12, 24, 36, 48 . . . . 108) and choose the smallest multiple that is exactly divisible by 54 and 12, i.e., 108.

If the LCM of 12 and 54 is 108, Find its GCF.

LCM(12, 54) × GCF(12, 54) = 12 × 54
Since the LCM of 12 and 54 = 108
⇒ 108 × GCF(12, 54) = 648
Therefore, the greatest common factor (GCF) = 648/108 = 6.

What is the Relation Between GCF and LCM of 54, 12?

The following equation can be used to express the relation between GCF and LCM of 54 and 12, i.e. GCF × LCM = 54 × 12.

Which of the following is the LCM of 54 and 12? 12, 27, 2, 108

The value of LCM of 54, 12 is the smallest common multiple of 54 and 12. The number satisfying the given condition is 108.

What is the Least Perfect Square Divisible by 54 and 12?

The least number divisible by 54 and 12 = LCM(54, 12)
LCM of 54 and 12 = 2 × 2 × 3 × 3 × 3 [Incomplete pair(s): 3]
⇒ Least perfect square divisible by each 54 and 12 = LCM(54, 12) × 3 = 324 [Square root of 324 = √324 = ±18]
Therefore, 324 is the required number.

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