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LCM of 6 and 27

LCM of 6 and 27 is the smallest number among all common multiples of 6 and 27. The first few multiples of 6 and 27 are (6, 12, 18, 24, . . . ) and (27, 54, 81, 108, . . . ) respectively. There are 3 commonly used methods to find LCM of 6 and 27 - by division method, by prime factorization, and by listing multiples.

1.	LCM of 6 and 27
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 6 and 27?

Answer: LCM of 6 and 27 is 54.

Explanation:

The LCM of two non-zero integers, x(6) and y(27), is the smallest positive integer m(54) that is divisible by both x(6) and y(27) without any remainder.

Methods to Find LCM of 6 and 27

Let's look at the different methods for finding the LCM of 6 and 27.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 6 and 27 by Listing Multiples

To calculate the LCM of 6 and 27 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 6 (6, 12, 18, 24, . . . ) and 27 (27, 54, 81, 108, . . . . )
Step 2: The common multiples from the multiples of 6 and 27 are 54, 108, . . .
Step 3: The smallest common multiple of 6 and 27 is 54.

∴ The least common multiple of 6 and 27 = 54.

LCM of 6 and 27 by Prime Factorization

Prime factorization of 6 and 27 is (2 × 3) = 2¹ × 3¹ and (3 × 3 × 3) = 3³ respectively. LCM of 6 and 27 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3³ = 54.
Hence, the LCM of 6 and 27 by prime factorization is 54.

LCM of 6 and 27 by Division Method

To calculate the LCM of 6 and 27 by the division method, we will divide the numbers(6, 27) by their prime factors (preferably common). The product of these divisors gives the LCM of 6 and 27.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6 and 27. Write this prime number(2) on the left of the given numbers(6 and 27), separated as per the ladder arrangement.
Step 2: If any of the given numbers (6, 27) 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 6 and 27 is the product of all prime numbers on the left, i.e. LCM(6, 27) by division method = 2 × 3 × 3 × 3 = 54.

☛ Also Check:

LCM of 6 and 27 Examples

Example 1: Verify the relationship between GCF and LCM of 6 and 27.

Solution:

The relation between GCF and LCM of 6 and 27 is given as,
LCM(6, 27) × GCF(6, 27) = Product of 6, 27
Prime factorization of 6 and 27 is given as, 6 = (2 × 3) = 2¹ × 3¹ and 27 = (3 × 3 × 3) = 3³
LCM(6, 27) = 54
GCF(6, 27) = 3
LHS = LCM(6, 27) × GCF(6, 27) = 54 × 3 = 162
RHS = Product of 6, 27 = 6 × 27 = 162
⇒ LHS = RHS = 162
Hence, verified.
Example 2: The GCD and LCM of two numbers are 3 and 54 respectively. If one number is 27, find the other number.

Solution:

Let the other number be b.
∵ GCD × LCM = 27 × b
⇒ b = (GCD × LCM)/27
⇒ b = (3 × 54)/27
⇒ b = 6
Therefore, the other number is 6.
Example 3: The product of two numbers is 162. If their GCD is 3, what is their LCM?

Solution:

Given: GCD = 3
product of numbers = 162
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 162/3
Therefore, the LCM is 54.
The probable combination for the given case is LCM(6, 27) = 54.

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FAQs on LCM of 6 and 27

What is the LCM of 6 and 27?

The LCM of 6 and 27 is 54. To find the LCM of 6 and 27, we need to find the multiples of 6 and 27 (multiples of 6 = 6, 12, 18, 24 . . . . 54; multiples of 27 = 27, 54, 81, 108) and choose the smallest multiple that is exactly divisible by 6 and 27, i.e., 54.

What are the Methods to Find LCM of 6 and 27?

The commonly used methods to find the LCM of 6 and 27 are:

Listing Multiples
Prime Factorization Method
Division Method

What is the Relation Between GCF and LCM of 6, 27?

The following equation can be used to express the relation between GCF and LCM of 6 and 27, i.e. GCF × LCM = 6 × 27.

What is the Least Perfect Square Divisible by 6 and 27?

The least number divisible by 6 and 27 = LCM(6, 27)
LCM of 6 and 27 = 2 × 3 × 3 × 3 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 6 and 27 = LCM(6, 27) × 2 × 3 = 324 [Square root of 324 = √324 = ±18]
Therefore, 324 is the required number.

If the LCM of 27 and 6 is 54, Find its GCF.

LCM(27, 6) × GCF(27, 6) = 27 × 6
Since the LCM of 27 and 6 = 54
⇒ 54 × GCF(27, 6) = 162
Therefore, the GCF = 162/54 = 3.

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