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LCM of 24, 36, and 72

LCM of 24, 36, and 72 is the smallest number among all common multiples of 24, 36, and 72. The first few multiples of 24, 36, and 72 are (24, 48, 72, 96, 120 . . .), (36, 72, 108, 144, 180 . . .), and (72, 144, 216, 288, 360 . . .) respectively. There are 3 commonly used methods to find LCM of 24, 36, 72 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 24, 36, and 72
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 24, 36, and 72?

Answer: LCM of 24, 36, and 72 is 72.

Explanation:

The LCM of three non-zero integers, a(24), b(36), and c(72), is the smallest positive integer m(72) that is divisible by a(24), b(36), and c(72) without any remainder.

Methods to Find LCM of 24, 36, and 72

Let's look at the different methods for finding the LCM of 24, 36, and 72.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 24, 36, and 72 by Prime Factorization

Prime factorization of 24, 36, and 72 is (2 × 2 × 2 × 3) = 2³ × 3¹, (2 × 2 × 3 × 3) = 2² × 3², and (2 × 2 × 2 × 3 × 3) = 2³ × 3² respectively. LCM of 24, 36, and 72 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² = 72.
Hence, the LCM of 24, 36, and 72 by prime factorization is 72.

LCM of 24, 36, and 72 by Listing Multiples

To calculate the LCM of 24, 36, 72 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 24 (24, 48, 72, 96, 120 . . .), 36 (36, 72, 108, 144, 180 . . .), and 72 (72, 144, 216, 288, 360 . . .).
Step 2: The common multiples from the multiples of 24, 36, and 72 are 72, 144, . . .
Step 3: The smallest common multiple of 24, 36, and 72 is 72.

∴ The least common multiple of 24, 36, and 72 = 72.

LCM of 24, 36, and 72 by Division Method

To calculate the LCM of 24, 36, and 72 by the division method, we will divide the numbers(24, 36, 72) by their prime factors (preferably common). The product of these divisors gives the LCM of 24, 36, and 72.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 24, 36, and 72. Write this prime number(2) on the left of the given numbers(24, 36, and 72), separated as per the ladder arrangement.
Step 2: If any of the given numbers (24, 36, 72) 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 24, 36, and 72 is the product of all prime numbers on the left, i.e. LCM(24, 36, 72) by division method = 2 × 2 × 2 × 3 × 3 = 72.

☛ Also Check:

LCM of 24, 36, and 72 Examples

Example 1: Calculate the LCM of 24, 36, and 72 using the GCD of the given numbers.

Solution:

Prime factorization of 24, 36, 72:
- 24 = 2³ × 3¹
- 36 = 2² × 3²
- 72 = 2³ × 3²
Therefore, GCD(24, 36) = 12, GCD(36, 72) = 36, GCD(24, 72) = 24, GCD(24, 36, 72) = 12
We know,
LCM(24, 36, 72) = [(24 × 36 × 72) × GCD(24, 36, 72)]/[GCD(24, 36) × GCD(36, 72) × GCD(24, 72)]
LCM(24, 36, 72) = (62208 × 12)/(12 × 36 × 24) = 72
⇒LCM(24, 36, 72) = 72
Example 2: Verify the relationship between the GCD and LCM of 24, 36, and 72.

Solution:

The relation between GCD and LCM of 24, 36, and 72 is given as,
LCM(24, 36, 72) = [(24 × 36 × 72) × GCD(24, 36, 72)]/[GCD(24, 36) × GCD(36, 72) × GCD(24, 72)]
⇒ Prime factorization of 24, 36 and 72:
- 24 = 2³ × 3¹
- 36 = 2² × 3²
- 72 = 2³ × 3²
∴ GCD of (24, 36), (36, 72), (24, 72) and (24, 36, 72) = 12, 36, 24 and 12 respectively.
Now, LHS = LCM(24, 36, 72) = 72.
And, RHS = [(24 × 36 × 72) × GCD(24, 36, 72)]/[GCD(24, 36) × GCD(36, 72) × GCD(24, 72)] = [(62208) × 12]/[12 × 36 × 24] = 72
LHS = RHS = 72.
Hence verified.
Example 3: Find the smallest number that is divisible by 24, 36, 72 exactly.

Solution:

The smallest number that is divisible by 24, 36, and 72 exactly is their LCM.
⇒ Multiples of 24, 36, and 72:
- Multiples of 24 = 24, 48, 72, 96, 120, 144, . . . .
- Multiples of 36 = 36, 72, 108, 144, 180, 216, . . . .
- Multiples of 72 = 72, 144, 216, 288, 360, 432, . . . .
Therefore, the LCM of 24, 36, and 72 is 72.

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FAQs on LCM of 24, 36, and 72

What is the LCM of 24, 36, and 72?

The LCM of 24, 36, and 72 is 72. To find the LCM (least common multiple) of 24, 36, and 72, we need to find the multiples of 24, 36, and 72 (multiples of 24 = 24, 48 . . . .; multiples of 36 = 36, 72 . . . .; multiples of 72 = 72, 144 . . . .) and choose the smallest multiple that is exactly divisible by 24, 36, and 72, i.e., 72.

Which of the following is the LCM of 24, 36, and 72? 18, 81, 11, 72

The value of LCM of 24, 36, 72 is the smallest common multiple of 24, 36, and 72. The number satisfying the given condition is 72.

How to Find the LCM of 24, 36, and 72 by Prime Factorization?

To find the LCM of 24, 36, and 72 using prime factorization, we will find the prime factors, (24 = 2³ × 3¹), (36 = 2² × 3²), and (72 = 2³ × 3²). LCM of 24, 36, and 72 is the product of prime factors raised to their respective highest exponent among the numbers 24, 36, and 72.
⇒ LCM of 24, 36, 72 = 2³ × 3² = 72.

What are the Methods to Find LCM of 24, 36, 72?

The commonly used methods to find the LCM of 24, 36, 72 are:

Listing Multiples
Prime Factorization Method
Division Method

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