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LCM of 24, 36, and 54

LCM of 24, 36, and 54 is the smallest number among all common multiples of 24, 36, and 54. The first few multiples of 24, 36, and 54 are (24, 48, 72, 96, 120 . . .), (36, 72, 108, 144, 180 . . .), and (54, 108, 162, 216, 270 . . .) respectively. There are 3 commonly used methods to find LCM of 24, 36, 54 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 24, 36, and 54
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 24, 36, and 54?

Answer: LCM of 24, 36, and 54 is 216.

Explanation:

The LCM of three non-zero integers, a(24), b(36), and c(54), is the smallest positive integer m(216) that is divisible by a(24), b(36), and c(54) without any remainder.

Methods to Find LCM of 24, 36, and 54

Let's look at the different methods for finding the LCM of 24, 36, and 54.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 24, 36, and 54 by Listing Multiples

To calculate the LCM of 24, 36, 54 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 24 (24, 48, 72, 96, 120 . . .), 36 (36, 72, 108, 144, 180 . . .), and 54 (54, 108, 162, 216, 270 . . .).
Step 2: The common multiples from the multiples of 24, 36, and 54 are 216, 432, . . .
Step 3: The smallest common multiple of 24, 36, and 54 is 216.

∴ The least common multiple of 24, 36, and 54 = 216.

LCM of 24, 36, and 54 by Prime Factorization

Prime factorization of 24, 36, and 54 is (2 × 2 × 2 × 3) = 2³ × 3¹, (2 × 2 × 3 × 3) = 2² × 3², and (2 × 3 × 3 × 3) = 2¹ × 3³ respectively. LCM of 24, 36, and 54 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3³ = 216.
Hence, the LCM of 24, 36, and 54 by prime factorization is 216.

LCM of 24, 36, and 54 by Division Method

To calculate the LCM of 24, 36, and 54 by the division method, we will divide the numbers(24, 36, 54) by their prime factors (preferably common). The product of these divisors gives the LCM of 24, 36, and 54.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 24, 36, and 54. Write this prime number(2) on the left of the given numbers(24, 36, and 54), separated as per the ladder arrangement.
Step 2: If any of the given numbers (24, 36, 54) 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 24, 36, and 54 is the product of all prime numbers on the left, i.e. LCM(24, 36, 54) by division method = 2 × 2 × 2 × 3 × 3 × 3 = 216.

☛ Also Check:

LCM of 24, 36, and 54 Examples

Example 1: Find the smallest number that is divisible by 24, 36, 54 exactly.

Solution:

The smallest number that is divisible by 24, 36, and 54 exactly is their LCM.
⇒ Multiples of 24, 36, and 54:
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, 192, 216, . . . .
- Multiples of 36 = 36, 72, 108, 144, 180, 216, . . . .
- Multiples of 54 = 54, 108, 162, 216, 270, . . . .
Therefore, the LCM of 24, 36, and 54 is 216.
Example 2: Calculate the LCM of 24, 36, and 54 using the GCD of the given numbers.

Solution:

Prime factorization of 24, 36, 54:
- 24 = 2³ × 3¹
- 36 = 2² × 3²
- 54 = 2¹ × 3³
Therefore, GCD(24, 36) = 12, GCD(36, 54) = 18, GCD(24, 54) = 6, GCD(24, 36, 54) = 6
We know,
LCM(24, 36, 54) = [(24 × 36 × 54) × GCD(24, 36, 54)]/[GCD(24, 36) × GCD(36, 54) × GCD(24, 54)]
LCM(24, 36, 54) = (46656 × 6)/(12 × 18 × 6) = 216
⇒LCM(24, 36, 54) = 216
Example 3: Verify the relationship between the GCD and LCM of 24, 36, and 54.

Solution:

The relation between GCD and LCM of 24, 36, and 54 is given as,
LCM(24, 36, 54) = [(24 × 36 × 54) × GCD(24, 36, 54)]/[GCD(24, 36) × GCD(36, 54) × GCD(24, 54)]
⇒ Prime factorization of 24, 36 and 54:
- 24 = 2³ × 3¹
- 36 = 2² × 3²
- 54 = 2¹ × 3³
∴ GCD of (24, 36), (36, 54), (24, 54) and (24, 36, 54) = 12, 18, 6 and 6 respectively.
Now, LHS = LCM(24, 36, 54) = 216.
And, RHS = [(24 × 36 × 54) × GCD(24, 36, 54)]/[GCD(24, 36) × GCD(36, 54) × GCD(24, 54)] = [(46656) × 6]/[12 × 18 × 6] = 216
LHS = RHS = 216.
Hence verified.

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FAQs on LCM of 24, 36, and 54

What is the LCM of 24, 36, and 54?

The LCM of 24, 36, and 54 is 216. To find the least common multiple (LCM) of 24, 36, and 54, we need to find the multiples of 24, 36, and 54 (multiples of 24 = 24, 48, 72, 96 . . . . 216 . . . . ; multiples of 36 = 36, 72, 108, 144, 216 . . . .; multiples of 54 = 54, 108, 162, 216 . . . .) and choose the smallest multiple that is exactly divisible by 24, 36, and 54, i.e., 216.

How to Find the LCM of 24, 36, and 54 by Prime Factorization?

To find the LCM of 24, 36, and 54 using prime factorization, we will find the prime factors, (24 = 2³ × 3¹), (36 = 2² × 3²), and (54 = 2¹ × 3³). LCM of 24, 36, and 54 is the product of prime factors raised to their respective highest exponent among the numbers 24, 36, and 54.
⇒ LCM of 24, 36, 54 = 2³ × 3³ = 216.

Which of the following is the LCM of 24, 36, and 54? 120, 100, 216, 35

The value of LCM of 24, 36, 54 is the smallest common multiple of 24, 36, and 54. The number satisfying the given condition is 216.

What is the Least Perfect Square Divisible by 24, 36, and 54?

The least number divisible by 24, 36, and 54 = LCM(24, 36, 54)
LCM of 24, 36, and 54 = 2 × 2 × 2 × 3 × 3 × 3 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 24, 36, and 54 = LCM(24, 36, 54) × 2 × 3 = 1296 [Square root of 1296 = √1296 = ±36]
Therefore, 1296 is the required number.

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