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LCM of 72, 108, and 2100

LCM of 72, 108, and 2100 is the smallest number among all common multiples of 72, 108, and 2100. The first few multiples of 72, 108, and 2100 are (72, 144, 216, 288, 360 . . .), (108, 216, 324, 432, 540 . . .), and (2100, 4200, 6300, 8400, 10500 . . .) respectively. There are 3 commonly used methods to find LCM of 72, 108, 2100 - by division method, by prime factorization, and by listing multiples.

1.	LCM of 72, 108, and 2100
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 72, 108, and 2100?

Answer: LCM of 72, 108, and 2100 is 37800.

Explanation:

The LCM of three non-zero integers, a(72), b(108), and c(2100), is the smallest positive integer m(37800) that is divisible by a(72), b(108), and c(2100) without any remainder.

Methods to Find LCM of 72, 108, and 2100

Let's look at the different methods for finding the LCM of 72, 108, and 2100.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 72, 108, and 2100 by Prime Factorization

Prime factorization of 72, 108, and 2100 is (2 × 2 × 2 × 3 × 3) = 2³ × 3², (2 × 2 × 3 × 3 × 3) = 2² × 3³, and (2 × 2 × 3 × 5 × 5 × 7) = 2² × 3¹ × 5² × 7¹ respectively. LCM of 72, 108, and 2100 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3³ × 5² × 7¹ = 37800.
Hence, the LCM of 72, 108, and 2100 by prime factorization is 37800.

LCM of 72, 108, and 2100 by Division Method

To calculate the LCM of 72, 108, and 2100 by the division method, we will divide the numbers(72, 108, 2100) by their prime factors (preferably common). The product of these divisors gives the LCM of 72, 108, and 2100.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 72, 108, and 2100. Write this prime number(2) on the left of the given numbers(72, 108, and 2100), separated as per the ladder arrangement.
Step 2: If any of the given numbers (72, 108, 2100) 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 72, 108, and 2100 is the product of all prime numbers on the left, i.e. LCM(72, 108, 2100) by division method = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 7 = 37800.

LCM of 72, 108, and 2100 by Listing Multiples

To calculate the LCM of 72, 108, 2100 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 72 (72, 144, 216, 288, 360 . . .), 108 (108, 216, 324, 432, 540 . . .), and 2100 (2100, 4200, 6300, 8400, 10500 . . .).
Step 2: The common multiples from the multiples of 72, 108, and 2100 are 37800, 75600, . . .
Step 3: The smallest common multiple of 72, 108, and 2100 is 37800.

∴ The least common multiple of 72, 108, and 2100 = 37800.

☛ Also Check:

LCM of 72, 108, and 2100 Examples

Example 1: Verify the relationship between the GCD and LCM of 72, 108, and 2100.

Solution:

The relation between GCD and LCM of 72, 108, and 2100 is given as,
LCM(72, 108, 2100) = [(72 × 108 × 2100) × GCD(72, 108, 2100)]/[GCD(72, 108) × GCD(108, 2100) × GCD(72, 2100)]
⇒ Prime factorization of 72, 108 and 2100:
- 72 = 2³ × 3²
- 108 = 2² × 3³
- 2100 = 2² × 3¹ × 5² × 7¹
∴ GCD of (72, 108), (108, 2100), (72, 2100) and (72, 108, 2100) = 36, 12, 12 and 12 respectively.
Now, LHS = LCM(72, 108, 2100) = 37800.
And, RHS = [(72 × 108 × 2100) × GCD(72, 108, 2100)]/[GCD(72, 108) × GCD(108, 2100) × GCD(72, 2100)] = [(16329600) × 12]/[36 × 12 × 12] = 37800
LHS = RHS = 37800.
Hence verified.
Example 2: Calculate the LCM of 72, 108, and 2100 using the GCD of the given numbers.

Solution:

Prime factorization of 72, 108, 2100:
- 72 = 2³ × 3²
- 108 = 2² × 3³
- 2100 = 2² × 3¹ × 5² × 7¹
Therefore, GCD(72, 108) = 36, GCD(108, 2100) = 12, GCD(72, 2100) = 12, GCD(72, 108, 2100) = 12
We know,
LCM(72, 108, 2100) = [(72 × 108 × 2100) × GCD(72, 108, 2100)]/[GCD(72, 108) × GCD(108, 2100) × GCD(72, 2100)]
LCM(72, 108, 2100) = (16329600 × 12)/(36 × 12 × 12) = 37800
⇒LCM(72, 108, 2100) = 37800
Example 3: Find the smallest number that is divisible by 72, 108, 2100 exactly.

Solution:

The value of LCM(72, 108, 2100) will be the smallest number that is exactly divisible by 72, 108, and 2100.
⇒ Multiples of 72, 108, and 2100:
- Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, 576, 648, 720, . . . ., 37656, 37728, 37800, . . . .
- Multiples of 108 = 108, 216, 324, 432, 540, 648, 756, 864, 972, 1080, . . . ., 37476, 37584, 37692, 37800, . . . .
- Multiples of 2100 = 2100, 4200, 6300, 8400, 10500, 12600, 14700, 16800, 18900, 21000, . . . ., 29400, 31500, 33600, 35700, 37800, . . . .
Therefore, the LCM of 72, 108, and 2100 is 37800.

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FAQs on LCM of 72, 108, and 2100

What is the LCM of 72, 108, and 2100?

The LCM of 72, 108, and 2100 is 37800. To find the LCM (least common multiple) of 72, 108, and 2100, we need to find the multiples of 72, 108, and 2100 (multiples of 72 = 72, 144, 216, 288 . . . . 37800 . . . . ; multiples of 108 = 108, 216, 324, 432 . . . . 37800 . . . . ; multiples of 2100 = 2100, 4200, 6300, 8400 . . . . 37800 . . . . ) and choose the smallest multiple that is exactly divisible by 72, 108, and 2100, i.e., 37800.

What are the Methods to Find LCM of 72, 108, 2100?

The commonly used methods to find the LCM of 72, 108, 2100 are:

Prime Factorization Method
Listing Multiples
Division Method

What is the Least Perfect Square Divisible by 72, 108, and 2100?

The least number divisible by 72, 108, and 2100 = LCM(72, 108, 2100)
LCM of 72, 108, and 2100 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 7 [Incomplete pair(s): 2, 3, 7]
⇒ Least perfect square divisible by each 72, 108, and 2100 = LCM(72, 108, 2100) × 2 × 3 × 7 = 1587600 [Square root of 1587600 = √1587600 = ±1260]
Therefore, 1587600 is the required number.

What is the Relation Between GCF and LCM of 72, 108, 2100?

The following equation can be used to express the relation between GCF and LCM of 72, 108, 2100, i.e. LCM(72, 108, 2100) = [(72 × 108 × 2100) × GCF(72, 108, 2100)]/[GCF(72, 108) × GCF(108, 2100) × GCF(72, 2100)].

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