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LCM of 30, 72, and 432

LCM of 30, 72, and 432 is the smallest number among all common multiples of 30, 72, and 432. The first few multiples of 30, 72, and 432 are (30, 60, 90, 120, 150 . . .), (72, 144, 216, 288, 360 . . .), and (432, 864, 1296, 1728, 2160 . . .) respectively. There are 3 commonly used methods to find LCM of 30, 72, 432 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 30, 72, and 432
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 30, 72, and 432?

Answer: LCM of 30, 72, and 432 is 2160.

Explanation:

The LCM of three non-zero integers, a(30), b(72), and c(432), is the smallest positive integer m(2160) that is divisible by a(30), b(72), and c(432) without any remainder.

Methods to Find LCM of 30, 72, and 432

The methods to find the LCM of 30, 72, and 432 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 30, 72, and 432 by Listing Multiples

To calculate the LCM of 30, 72, 432 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 . . .), 72 (72, 144, 216, 288, 360 . . .), and 432 (432, 864, 1296, 1728, 2160 . . .).
Step 2: The common multiples from the multiples of 30, 72, and 432 are 2160, 4320, . . .
Step 3: The smallest common multiple of 30, 72, and 432 is 2160.

∴ The least common multiple of 30, 72, and 432 = 2160.

LCM of 30, 72, and 432 by Prime Factorization

Prime factorization of 30, 72, and 432 is (2 × 3 × 5) = 2¹ × 3¹ × 5¹, (2 × 2 × 2 × 3 × 3) = 2³ × 3², and (2 × 2 × 2 × 2 × 3 × 3 × 3) = 2⁴ × 3³ respectively. LCM of 30, 72, and 432 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3³ × 5¹ = 2160.
Hence, the LCM of 30, 72, and 432 by prime factorization is 2160.

LCM of 30, 72, and 432 by Division Method

To calculate the LCM of 30, 72, and 432 by the division method, we will divide the numbers(30, 72, 432) by their prime factors (preferably common). The product of these divisors gives the LCM of 30, 72, and 432.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 30, 72, and 432. Write this prime number(2) on the left of the given numbers(30, 72, and 432), separated as per the ladder arrangement.
Step 2: If any of the given numbers (30, 72, 432) 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, 72, and 432 is the product of all prime numbers on the left, i.e. LCM(30, 72, 432) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 = 2160.

☛ Also Check:

LCM of 30, 72, and 432 Examples

Example 1: Calculate the LCM of 30, 72, and 432 using the GCD of the given numbers.

Solution:

Prime factorization of 30, 72, 432:
- 30 = 2¹ × 3¹ × 5¹
- 72 = 2³ × 3²
- 432 = 2⁴ × 3³
Therefore, GCD(30, 72) = 6, GCD(72, 432) = 72, GCD(30, 432) = 6, GCD(30, 72, 432) = 6
We know,
LCM(30, 72, 432) = [(30 × 72 × 432) × GCD(30, 72, 432)]/[GCD(30, 72) × GCD(72, 432) × GCD(30, 432)]
LCM(30, 72, 432) = (933120 × 6)/(6 × 72 × 6) = 2160
⇒LCM(30, 72, 432) = 2160
Example 2: Verify the relationship between the GCD and LCM of 30, 72, and 432.

Solution:

The relation between GCD and LCM of 30, 72, and 432 is given as,
LCM(30, 72, 432) = [(30 × 72 × 432) × GCD(30, 72, 432)]/[GCD(30, 72) × GCD(72, 432) × GCD(30, 432)]
⇒ Prime factorization of 30, 72 and 432:
- 30 = 2¹ × 3¹ × 5¹
- 72 = 2³ × 3²
- 432 = 2⁴ × 3³
∴ GCD of (30, 72), (72, 432), (30, 432) and (30, 72, 432) = 6, 72, 6 and 6 respectively.
Now, LHS = LCM(30, 72, 432) = 2160.
And, RHS = [(30 × 72 × 432) × GCD(30, 72, 432)]/[GCD(30, 72) × GCD(72, 432) × GCD(30, 432)] = [(933120) × 6]/[6 × 72 × 6] = 2160
LHS = RHS = 2160.
Hence verified.
Example 3: Find the smallest number that is divisible by 30, 72, 432 exactly.

Solution:

The value of LCM(30, 72, 432) will be the smallest number that is exactly divisible by 30, 72, and 432.
⇒ Multiples of 30, 72, and 432:
- Multiples of 30 = 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, . . . ., 2040, 2070, 2100, 2130, 2160, . . . .
- Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, 576, 648, 720, . . . ., 1944, 2016, 2088, 2160, . . . .
- Multiples of 432 = 432, 864, 1296, 1728, 2160, 2592, 3024, 3456, 3888, 4320, . . . ., 1296, 1728, 2160, . . . .
Therefore, the LCM of 30, 72, and 432 is 2160.

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FAQs on LCM of 30, 72, and 432

What is the LCM of 30, 72, and 432?

The LCM of 30, 72, and 432 is 2160. To find the LCM (least common multiple) of 30, 72, and 432, we need to find the multiples of 30, 72, and 432 (multiples of 30 = 30, 60, 90, 120 . . . . 2160 . . . . ; multiples of 72 = 72, 144, 216, 288 . . . . 2160 . . . . ; multiples of 432 = 432, 864, 1296, 1728 . . . . 2160 . . . . ) and choose the smallest multiple that is exactly divisible by 30, 72, and 432, i.e., 2160.

Which of the following is the LCM of 30, 72, and 432? 45, 2160, 28, 32

The value of LCM of 30, 72, 432 is the smallest common multiple of 30, 72, and 432. The number satisfying the given condition is 2160.

What is the Least Perfect Square Divisible by 30, 72, and 432?

The least number divisible by 30, 72, and 432 = LCM(30, 72, 432)
LCM of 30, 72, and 432 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 [Incomplete pair(s): 3, 5]
⇒ Least perfect square divisible by each 30, 72, and 432 = LCM(30, 72, 432) × 3 × 5 = 32400 [Square root of 32400 = √32400 = ±180]
Therefore, 32400 is the required number.

What are the Methods to Find LCM of 30, 72, 432?

The commonly used methods to find the LCM of 30, 72, 432 are:

Listing Multiples
Prime Factorization Method
Division Method

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