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LCM of 12, 15, 20, and 54

LCM of 12, 15, 20, and 54 is the smallest number among all common multiples of 12, 15, 20, and 54. The first few multiples of 12, 15, 20, and 54 are (12, 24, 36, 48, 60 . . .), (15, 30, 45, 60, 75 . . .), (20, 40, 60, 80, 100 . . .), and (54, 108, 162, 216, 270 . . .) respectively. There are 3 commonly used methods to find LCM of 12, 15, 20, 54 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 12, 15, 20, and 54
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12, 15, 20, and 54?

Answer: LCM of 12, 15, 20, and 54 is 540.

Explanation:

The LCM of four non-zero integers, a(12), b(15), c(20), and d(54), is the smallest positive integer m(540) that is divisible by a(12), b(15), c(20), and d(54) without any remainder.

Methods to Find LCM of 12, 15, 20, and 54

The methods to find the LCM of 12, 15, 20, and 54 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 12, 15, 20, and 54 by Listing Multiples

To calculate the LCM of 12, 15, 20, 54 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 12 (12, 24, 36, 48, 60 . . .), 15 (15, 30, 45, 60, 75 . . .), 20 (20, 40, 60, 80, 100 . . .), and 54 (54, 108, 162, 216, 270 . . .).
Step 2: The common multiples from the multiples of 12, 15, 20, and 54 are 540, 1080, . . .
Step 3: The smallest common multiple of 12, 15, 20, and 54 is 540.

∴ The least common multiple of 12, 15, 20, and 54 = 540.

LCM of 12, 15, 20, and 54 by Prime Factorization

Prime factorization of 12, 15, 20, and 54 is (2 × 2 × 3) = 2² × 3¹, (3 × 5) = 3¹ × 5¹, (2 × 2 × 5) = 2² × 5¹, and (2 × 3 × 3 × 3) = 2¹ × 3³ respectively. LCM of 12, 15, 20, and 54 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3³ × 5¹ = 540.
Hence, the LCM of 12, 15, 20, and 54 by prime factorization is 540.

LCM of 12, 15, 20, and 54 by Division Method

To calculate the LCM of 12, 15, 20, and 54 by the division method, we will divide the numbers(12, 15, 20, 54) by their prime factors (preferably common). The product of these divisors gives the LCM of 12, 15, 20, and 54.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 15, 20, and 54. Write this prime number(2) on the left of the given numbers(12, 15, 20, and 54), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 15, 20, 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 12, 15, 20, and 54 is the product of all prime numbers on the left, i.e. LCM(12, 15, 20, 54) by division method = 2 × 2 × 3 × 3 × 3 × 5 = 540.

☛ Also Check:

LCM of 12, 15, 20, and 54 Examples

Example 1: Find the smallest number that is divisible by 12, 15, 20, 54 exactly.

Solution:

The value of LCM(12, 15, 20, 54) will be the smallest number that is exactly divisible by 12, 15, 20, and 54.
⇒ Multiples of 12, 15, 20, and 54:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . ., 492, 504, 516, 528, 540, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, . . . ., 495, 510, 525, 540, . . . .
- Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, . . . ., 460, 480, 500, 520, 540, . . . .
- Multiples of 54 = 54, 108, 162, 216, 270, 324, 378, 432, 486, 540, . . . ., 432, 486, 540, . . . .
Therefore, the LCM of 12, 15, 20, and 54 is 540.
Example 2: Find the smallest number which when divided by 12, 15, 20, and 54 leaves 3 as the remainder in each case.

Solution:

The smallest number exactly divisible by 12, 15, 20, and 54 = LCM(12, 15, 20, 54) ⇒ Smallest number which leaves 3 as remainder when divided by 12, 15, 20, and 54 = LCM(12, 15, 20, 54) + 3
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 20 = 2² × 5¹
- 54 = 2¹ × 3³
LCM(12, 15, 20, 54) = 2² × 3³ × 5¹ = 540
⇒ The required number = 540 + 3 = 543.
Example 3: Which of the following is the LCM of 12, 15, 20, 54? 81, 110, 540, 105.

Solution:

The value of LCM of 12, 15, 20, and 54 is the smallest common multiple of 12, 15, 20, and 54. The number satisfying the given condition is 540. ∴LCM(12, 15, 20, 54) = 540.

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FAQs on LCM of 12, 15, 20, and 54

What is the LCM of 12, 15, 20, and 54?

The LCM of 12, 15, 20, and 54 is 540. To find the LCM of 12, 15, 20, and 54, we need to find the multiples of 12, 15, 20, and 54 (multiples of 12 = 12, 24, 36, 48 . . . . 540 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 540 . . . . ; multiples of 20 = 20, 40, 60, 80 . . . . 540 . . . . ; multiples of 54 = 54, 108, 162, 216 . . . . 540 . . . . ) and choose the smallest multiple that is exactly divisible by 12, 15, 20, and 54, i.e., 540.

What are the Methods to Find LCM of 12, 15, 20, 54?

The commonly used methods to find the LCM of 12, 15, 20, 54 are:

Prime Factorization Method
Division Method
Listing Multiples

How to Find the LCM of 12, 15, 20, and 54 by Prime Factorization?

To find the LCM of 12, 15, 20, and 54 using prime factorization, we will find the prime factors, (12 = 2² × 3¹), (15 = 3¹ × 5¹), (20 = 2² × 5¹), and (54 = 2¹ × 3³). LCM of 12, 15, 20, and 54 is the product of prime factors raised to their respective highest exponent among the numbers 12, 15, 20, and 54.
⇒ LCM of 12, 15, 20, 54 = 2² × 3³ × 5¹ = 540.

Which of the following is the LCM of 12, 15, 20, and 54? 12, 105, 540, 10

The value of LCM of 12, 15, 20, 54 is the smallest common multiple of 12, 15, 20, and 54. The number satisfying the given condition is 540.

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