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LCM of 12, 15, and 18

LCM of 12, 15, and 18 is the smallest number among all common multiples of 12, 15, and 18. The first few multiples of 12, 15, and 18 are (12, 24, 36, 48, 60 . . .), (15, 30, 45, 60, 75 . . .), and (18, 36, 54, 72, 90 . . .) respectively. There are 3 commonly used methods to find LCM of 12, 15, 18 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 12, 15, and 18
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12, 15, and 18?

Answer: LCM of 12, 15, and 18 is 180.

Explanation:

The LCM of three non-zero integers, a(12), b(15), and c(18), is the smallest positive integer m(180) that is divisible by a(12), b(15), and c(18) without any remainder.

Methods to Find LCM of 12, 15, and 18

The methods to find the LCM of 12, 15, and 18 are explained below.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 12, 15, and 18 by Division Method

To calculate the LCM of 12, 15, and 18 by the division method, we will divide the numbers(12, 15, 18) by their prime factors (preferably common). The product of these divisors gives the LCM of 12, 15, and 18.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 15, and 18. Write this prime number(2) on the left of the given numbers(12, 15, and 18), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 15, 18) 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 18 is the product of all prime numbers on the left, i.e. LCM(12, 15, 18) by division method = 2 × 2 × 3 × 3 × 5 = 180.

LCM of 12, 15, and 18 by Prime Factorization

Prime factorization of 12, 15, and 18 is (2 × 2 × 3) = 2² × 3¹, (3 × 5) = 3¹ × 5¹, and (2 × 3 × 3) = 2¹ × 3² respectively. LCM of 12, 15, and 18 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 5¹ = 180.
Hence, the LCM of 12, 15, and 18 by prime factorization is 180.

LCM of 12, 15, and 18 by Listing Multiples

To calculate the LCM of 12, 15, 18 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 18 (18, 36, 54, 72, 90 . . .).
Step 2: The common multiples from the multiples of 12, 15, and 18 are 180, 360, . . .
Step 3: The smallest common multiple of 12, 15, and 18 is 180.

∴ The least common multiple of 12, 15, and 18 = 180.

☛ Also Check:

LCM of 12, 15, and 18 Examples

Example 1: Find the smallest number that is divisible by 12, 15, 18 exactly.

Solution:

The smallest number that is divisible by 12, 15, and 18 exactly is their LCM.
⇒ Multiples of 12, 15, and 18:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, . . . .
- Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, . . . .
Therefore, the LCM of 12, 15, and 18 is 180.
Example 2: Calculate the LCM of 12, 15, and 18 using the GCD of the given numbers.

Solution:

Prime factorization of 12, 15, 18:
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 18 = 2¹ × 3²
Therefore, GCD(12, 15) = 3, GCD(15, 18) = 3, GCD(12, 18) = 6, GCD(12, 15, 18) = 3
We know,
LCM(12, 15, 18) = [(12 × 15 × 18) × GCD(12, 15, 18)]/[GCD(12, 15) × GCD(15, 18) × GCD(12, 18)]
LCM(12, 15, 18) = (3240 × 3)/(3 × 3 × 6) = 180
⇒LCM(12, 15, 18) = 180
Example 3: Verify the relationship between the GCD and LCM of 12, 15, and 18.

Solution:

The relation between GCD and LCM of 12, 15, and 18 is given as,
LCM(12, 15, 18) = [(12 × 15 × 18) × GCD(12, 15, 18)]/[GCD(12, 15) × GCD(15, 18) × GCD(12, 18)]
⇒ Prime factorization of 12, 15 and 18:
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 18 = 2¹ × 3²
∴ GCD of (12, 15), (15, 18), (12, 18) and (12, 15, 18) = 3, 3, 6 and 3 respectively.
Now, LHS = LCM(12, 15, 18) = 180.
And, RHS = [(12 × 15 × 18) × GCD(12, 15, 18)]/[GCD(12, 15) × GCD(15, 18) × GCD(12, 18)] = [(3240) × 3]/[3 × 3 × 6] = 180
LHS = RHS = 180.
Hence verified.

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FAQs on LCM of 12, 15, and 18

What is the LCM of 12, 15, and 18?

The LCM of 12, 15, and 18 is 180. To find the LCM (least common multiple) of 12, 15, and 18, we need to find the multiples of 12, 15, and 18 (multiples of 12 = 12, 24, 36, 48 . . . . 180 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 180 . . . . ; multiples of 18 = 18, 36, 54, 72 . . . . 180 . . . . ) and choose the smallest multiple that is exactly divisible by 12, 15, and 18, i.e., 180.

How to Find the LCM of 12, 15, and 18 by Prime Factorization?

To find the LCM of 12, 15, and 18 using prime factorization, we will find the prime factors, (12 = 2² × 3¹), (15 = 3¹ × 5¹), and (18 = 2¹ × 3²). LCM of 12, 15, and 18 is the product of prime factors raised to their respective highest exponent among the numbers 12, 15, and 18.
⇒ LCM of 12, 15, 18 = 2² × 3² × 5¹ = 180.

What are the Methods to Find LCM of 12, 15, 18?

The commonly used methods to find the LCM of 12, 15, 18 are:

Prime Factorization Method
Division Method
Listing Multiples

What is the Relation Between GCF and LCM of 12, 15, 18?

The following equation can be used to express the relation between GCF and LCM of 12, 15, 18, i.e. LCM(12, 15, 18) = [(12 × 15 × 18) × GCF(12, 15, 18)]/[GCF(12, 15) × GCF(15, 18) × GCF(12, 18)].

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