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LCM of 12, 15, and 45

LCM of 12, 15, and 45 is the smallest number among all common multiples of 12, 15, and 45. The first few multiples of 12, 15, and 45 are (12, 24, 36, 48, 60 . . .), (15, 30, 45, 60, 75 . . .), and (45, 90, 135, 180, 225 . . .) respectively. There are 3 commonly used methods to find LCM of 12, 15, 45 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 12, 15, and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12, 15, and 45?

Answer: LCM of 12, 15, and 45 is 180.

Explanation:

The LCM of three non-zero integers, a(12), b(15), and c(45), is the smallest positive integer m(180) that is divisible by a(12), b(15), and c(45) without any remainder.

Methods to Find LCM of 12, 15, and 45

The methods to find the LCM of 12, 15, and 45 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 12, 15, and 45 by Listing Multiples

To calculate the LCM of 12, 15, 45 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 45 (45, 90, 135, 180, 225 . . .).
Step 2: The common multiples from the multiples of 12, 15, and 45 are 180, 360, . . .
Step 3: The smallest common multiple of 12, 15, and 45 is 180.

∴ The least common multiple of 12, 15, and 45 = 180.

LCM of 12, 15, and 45 by Prime Factorization

Prime factorization of 12, 15, and 45 is (2 × 2 × 3) = 2² × 3¹, (3 × 5) = 3¹ × 5¹, and (3 × 3 × 5) = 3² × 5¹ respectively. LCM of 12, 15, and 45 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 45 by prime factorization is 180.

LCM of 12, 15, and 45 by Division Method

To calculate the LCM of 12, 15, and 45 by the division method, we will divide the numbers(12, 15, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 12, 15, and 45.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 15, and 45. Write this prime number(2) on the left of the given numbers(12, 15, and 45), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 15, 45) 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 45 is the product of all prime numbers on the left, i.e. LCM(12, 15, 45) by division method = 2 × 2 × 3 × 3 × 5 = 180.

☛ Also Check:

LCM of 12, 15, and 45 Examples

Example 1: Calculate the LCM of 12, 15, and 45 using the GCD of the given numbers.

Solution:

Prime factorization of 12, 15, 45:
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 45 = 3² × 5¹
Therefore, GCD(12, 15) = 3, GCD(15, 45) = 15, GCD(12, 45) = 3, GCD(12, 15, 45) = 3
We know,
LCM(12, 15, 45) = [(12 × 15 × 45) × GCD(12, 15, 45)]/[GCD(12, 15) × GCD(15, 45) × GCD(12, 45)]
LCM(12, 15, 45) = (8100 × 3)/(3 × 15 × 3) = 180
⇒LCM(12, 15, 45) = 180
Example 2: Find the smallest number that is divisible by 12, 15, 45 exactly.

Solution:

The smallest number that is divisible by 12, 15, and 45 exactly is their LCM.
⇒ Multiples of 12, 15, and 45:
- 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 45 = 45, 90, 135, 180, 225, 270, . . . .
Therefore, the LCM of 12, 15, and 45 is 180.
Example 3: Verify the relationship between the GCD and LCM of 12, 15, and 45.

Solution:

The relation between GCD and LCM of 12, 15, and 45 is given as,
LCM(12, 15, 45) = [(12 × 15 × 45) × GCD(12, 15, 45)]/[GCD(12, 15) × GCD(15, 45) × GCD(12, 45)]
⇒ Prime factorization of 12, 15 and 45:
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹
- 45 = 3² × 5¹
∴ GCD of (12, 15), (15, 45), (12, 45) and (12, 15, 45) = 3, 15, 3 and 3 respectively.
Now, LHS = LCM(12, 15, 45) = 180.
And, RHS = [(12 × 15 × 45) × GCD(12, 15, 45)]/[GCD(12, 15) × GCD(15, 45) × GCD(12, 45)] = [(8100) × 3]/[3 × 15 × 3] = 180
LHS = RHS = 180.
Hence verified.

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FAQs on LCM of 12, 15, and 45

What is the LCM of 12, 15, and 45?

The LCM of 12, 15, and 45 is 180. To find the LCM (least common multiple) of 12, 15, and 45, we need to find the multiples of 12, 15, and 45 (multiples of 12 = 12, 24, 36, 48 . . . . 180 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 180 . . . . ; multiples of 45 = 45, 90, 135, 180 . . . .) and choose the smallest multiple that is exactly divisible by 12, 15, and 45, i.e., 180.

What is the Relation Between GCF and LCM of 12, 15, 45?

The following equation can be used to express the relation between GCF and LCM of 12, 15, 45, i.e. LCM(12, 15, 45) = [(12 × 15 × 45) × GCF(12, 15, 45)]/[GCF(12, 15) × GCF(15, 45) × GCF(12, 45)].

What is the Least Perfect Square Divisible by 12, 15, and 45?

The least number divisible by 12, 15, and 45 = LCM(12, 15, 45)
LCM of 12, 15, and 45 = 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 5]
⇒ Least perfect square divisible by each 12, 15, and 45 = LCM(12, 15, 45) × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

What are the Methods to Find LCM of 12, 15, 45?

The commonly used methods to find the LCM of 12, 15, 45 are:

Division Method
Prime Factorization Method
Listing Multiples

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