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LCM of 30, 45, and 60

LCM of 30, 45, and 60 is the smallest number among all common multiples of 30, 45, and 60. The first few multiples of 30, 45, and 60 are (30, 60, 90, 120, 150 . . .), (45, 90, 135, 180, 225 . . .), and (60, 120, 180, 240, 300 . . .) respectively. There are 3 commonly used methods to find LCM of 30, 45, 60 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 30, 45, and 60
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 30, 45, and 60?

Answer: LCM of 30, 45, and 60 is 180.

Explanation:

The LCM of three non-zero integers, a(30), b(45), and c(60), is the smallest positive integer m(180) that is divisible by a(30), b(45), and c(60) without any remainder.

Methods to Find LCM of 30, 45, and 60

The methods to find the LCM of 30, 45, and 60 are explained below.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 30, 45, and 60 by Division Method

To calculate the LCM of 30, 45, and 60 by the division method, we will divide the numbers(30, 45, 60) by their prime factors (preferably common). The product of these divisors gives the LCM of 30, 45, and 60.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 30, 45, and 60. Write this prime number(2) on the left of the given numbers(30, 45, and 60), separated as per the ladder arrangement.
Step 2: If any of the given numbers (30, 45, 60) 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 30, 45, and 60 is the product of all prime numbers on the left, i.e. LCM(30, 45, 60) by division method = 2 × 2 × 3 × 3 × 5 = 180.

LCM of 30, 45, and 60 by Prime Factorization

Prime factorization of 30, 45, and 60 is (2 × 3 × 5) = 2¹ × 3¹ × 5¹, (3 × 3 × 5) = 3² × 5¹, and (2 × 2 × 3 × 5) = 2² × 3¹ × 5¹ respectively. LCM of 30, 45, and 60 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 5¹ = 180.
Hence, the LCM of 30, 45, and 60 by prime factorization is 180.

LCM of 30, 45, and 60 by Listing Multiples

To calculate the LCM of 30, 45, 60 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 30 (30, 60, 90, 120, 150 . . .), 45 (45, 90, 135, 180, 225 . . .), and 60 (60, 120, 180, 240, 300 . . .).
Step 2: The common multiples from the multiples of 30, 45, and 60 are 180, 360, . . .
Step 3: The smallest common multiple of 30, 45, and 60 is 180.

∴ The least common multiple of 30, 45, and 60 = 180.

☛ Also Check:

LCM of 30, 45, and 60 Examples

Example 1: Calculate the LCM of 30, 45, and 60 using the GCD of the given numbers.

Solution:

Prime factorization of 30, 45, 60:
- 30 = 2¹ × 3¹ × 5¹
- 45 = 3² × 5¹
- 60 = 2² × 3¹ × 5¹
Therefore, GCD(30, 45) = 15, GCD(45, 60) = 15, GCD(30, 60) = 30, GCD(30, 45, 60) = 15
We know,
LCM(30, 45, 60) = [(30 × 45 × 60) × GCD(30, 45, 60)]/[GCD(30, 45) × GCD(45, 60) × GCD(30, 60)]
LCM(30, 45, 60) = (81000 × 15)/(15 × 15 × 30) = 180
⇒LCM(30, 45, 60) = 180
Example 2: Verify the relationship between the GCD and LCM of 30, 45, and 60.

Solution:

The relation between GCD and LCM of 30, 45, and 60 is given as,
LCM(30, 45, 60) = [(30 × 45 × 60) × GCD(30, 45, 60)]/[GCD(30, 45) × GCD(45, 60) × GCD(30, 60)]
⇒ Prime factorization of 30, 45 and 60:
- 30 = 2¹ × 3¹ × 5¹
- 45 = 3² × 5¹
- 60 = 2² × 3¹ × 5¹
∴ GCD of (30, 45), (45, 60), (30, 60) and (30, 45, 60) = 15, 15, 30 and 15 respectively.
Now, LHS = LCM(30, 45, 60) = 180.
And, RHS = [(30 × 45 × 60) × GCD(30, 45, 60)]/[GCD(30, 45) × GCD(45, 60) × GCD(30, 60)] = [(81000) × 15]/[15 × 15 × 30] = 180
LHS = RHS = 180.
Hence verified.
Example 3: Find the smallest number that is divisible by 30, 45, 60 exactly.

Solution:

The smallest number that is divisible by 30, 45, and 60 exactly is their LCM.
⇒ Multiples of 30, 45, and 60:
- Multiples of 30 = 30, 60, 90, 120, 150, 180, 210, . . . .
- Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, . . . .
- Multiples of 60 = 60, 120, 180, 240, 300, 360, 420, . . . .
Therefore, the LCM of 30, 45, and 60 is 180.

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FAQs on LCM of 30, 45, and 60

What is the LCM of 30, 45, and 60?

The LCM of 30, 45, and 60 is 180. To find the LCM of 30, 45, and 60, we need to find the multiples of 30, 45, and 60 (multiples of 30 = 30, 60, 90, 120, 180 . . . .; multiples of 45 = 45, 90, 135, 180 . . . .; multiples of 60 = 60, 120, 180, 240 . . . .) and choose the smallest multiple that is exactly divisible by 30, 45, and 60, i.e., 180.

What is the Least Perfect Square Divisible by 30, 45, and 60?

The least number divisible by 30, 45, and 60 = LCM(30, 45, 60)
LCM of 30, 45, and 60 = 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 5]
⇒ Least perfect square divisible by each 30, 45, and 60 = LCM(30, 45, 60) × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

What are the Methods to Find LCM of 30, 45, 60?

The commonly used methods to find the LCM of 30, 45, 60 are:

Division Method
Listing Multiples
Prime Factorization Method

How to Find the LCM of 30, 45, and 60 by Prime Factorization?

To find the LCM of 30, 45, and 60 using prime factorization, we will find the prime factors, (30 = 2¹ × 3¹ × 5¹), (45 = 3² × 5¹), and (60 = 2² × 3¹ × 5¹). LCM of 30, 45, and 60 is the product of prime factors raised to their respective highest exponent among the numbers 30, 45, and 60.
⇒ LCM of 30, 45, 60 = 2² × 3² × 5¹ = 180.

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