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LCM of 12, 15, and 20

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

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

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

Answer: LCM of 12, 15, and 20 is 60.

Explanation:

The LCM of three non-zero integers, a(12), b(15), and c(20), is the smallest positive integer m(60) that is divisible by a(12), b(15), and c(20) without any remainder.

Methods to Find LCM of 12, 15, and 20

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

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 12, 15, and 20 by Listing Multiples

To calculate the LCM of 12, 15, 20 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 . . .), and 20 (20, 40, 60, 80, 100 . . .).
Step 2: The common multiples from the multiples of 12, 15, and 20 are 60, 120, . . .
Step 3: The smallest common multiple of 12, 15, and 20 is 60.

∴ The least common multiple of 12, 15, and 20 = 60.

LCM of 12, 15, and 20 by Division Method

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

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 15, and 20. Write this prime number(2) on the left of the given numbers(12, 15, and 20), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 15, 20) 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, and 20 is the product of all prime numbers on the left, i.e. LCM(12, 15, 20) by division method = 2 × 2 × 3 × 5 = 60.

LCM of 12, 15, and 20 by Prime Factorization

Prime factorization of 12, 15, and 20 is (2 × 2 × 3) = 2² × 3¹, (3 × 5) = 3¹ × 5¹, and (2 × 2 × 5) = 2² × 5¹ respectively. LCM of 12, 15, and 20 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3¹ × 5¹ = 60.
Hence, the LCM of 12, 15, and 20 by prime factorization is 60.

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LCM of 12, 15, and 20 Examples

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

Solution:

The smallest number that is divisible by 12, 15, and 20 exactly is their LCM.
⇒ Multiples of 12, 15, and 20:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, . . . .
- Multiples of 20 = 20, 40, 60, 80, 100, 120, . . . .
Therefore, the LCM of 12, 15, and 20 is 60.
Example 2: Calculate the LCM of 12, 15, and 20 using the GCD of the given numbers.

Solution:

Prime factorization of 12, 15, 20:
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 20 = 2² × 5¹
Therefore, GCD(12, 15) = 3, GCD(15, 20) = 5, GCD(12, 20) = 4, GCD(12, 15, 20) = 1
We know,
LCM(12, 15, 20) = [(12 × 15 × 20) × GCD(12, 15, 20)]/[GCD(12, 15) × GCD(15, 20) × GCD(12, 20)]
LCM(12, 15, 20) = (3600 × 1)/(3 × 5 × 4) = 60
⇒LCM(12, 15, 20) = 60
Example 3: Verify the relationship between the GCD and LCM of 12, 15, and 20.

Solution:

The relation between GCD and LCM of 12, 15, and 20 is given as,
LCM(12, 15, 20) = [(12 × 15 × 20) × GCD(12, 15, 20)]/[GCD(12, 15) × GCD(15, 20) × GCD(12, 20)]
⇒ Prime factorization of 12, 15 and 20:
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 20 = 2² × 5¹
∴ GCD of (12, 15), (15, 20), (12, 20) and (12, 15, 20) = 3, 5, 4 and 1 respectively.
Now, LHS = LCM(12, 15, 20) = 60.
And, RHS = [(12 × 15 × 20) × GCD(12, 15, 20)]/[GCD(12, 15) × GCD(15, 20) × GCD(12, 20)] = [(3600) × 1]/[3 × 5 × 4] = 60
LHS = RHS = 60.
Hence verified.

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FAQs on LCM of 12, 15, and 20

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

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

What is the Least Perfect Square Divisible by 12, 15, and 20?

The least number divisible by 12, 15, and 20 = LCM(12, 15, 20)
LCM of 12, 15, and 20 = 2 × 2 × 3 × 5 [Incomplete pair(s): 3, 5]
⇒ Least perfect square divisible by each 12, 15, and 20 = LCM(12, 15, 20) × 3 × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

Which of the following is the LCM of 12, 15, and 20? 32, 52, 60, 36

The value of LCM of 12, 15, 20 is the smallest common multiple of 12, 15, and 20. The number satisfying the given condition is 60.

What is the Relation Between GCF and LCM of 12, 15, 20?

The following equation can be used to express the relation between GCF and LCM of 12, 15, 20, i.e. LCM(12, 15, 20) = [(12 × 15 × 20) × GCF(12, 15, 20)]/[GCF(12, 15) × GCF(15, 20) × GCF(12, 20)].

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