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LCM of 24, 15, and 36

LCM of 24, 15, and 36 is the smallest number among all common multiples of 24, 15, and 36. The first few multiples of 24, 15, and 36 are (24, 48, 72, 96, 120 . . .), (15, 30, 45, 60, 75 . . .), and (36, 72, 108, 144, 180 . . .) respectively. There are 3 commonly used methods to find LCM of 24, 15, 36 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 24, 15, and 36
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 24, 15, and 36?

Answer: LCM of 24, 15, and 36 is 360.

Explanation:

The LCM of three non-zero integers, a(24), b(15), and c(36), is the smallest positive integer m(360) that is divisible by a(24), b(15), and c(36) without any remainder.

Methods to Find LCM of 24, 15, and 36

The methods to find the LCM of 24, 15, and 36 are explained below.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 24, 15, and 36 by Division Method

To calculate the LCM of 24, 15, and 36 by the division method, we will divide the numbers(24, 15, 36) by their prime factors (preferably common). The product of these divisors gives the LCM of 24, 15, and 36.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 24, 15, and 36. Write this prime number(2) on the left of the given numbers(24, 15, and 36), separated as per the ladder arrangement.
Step 2: If any of the given numbers (24, 15, 36) 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, 15, and 36 is the product of all prime numbers on the left, i.e. LCM(24, 15, 36) by division method = 2 × 2 × 2 × 3 × 3 × 5 = 360.

LCM of 24, 15, and 36 by Prime Factorization

Prime factorization of 24, 15, and 36 is (2 × 2 × 2 × 3) = 2³ × 3¹, (3 × 5) = 3¹ × 5¹, and (2 × 2 × 3 × 3) = 2² × 3² respectively. LCM of 24, 15, and 36 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ = 360.
Hence, the LCM of 24, 15, and 36 by prime factorization is 360.

LCM of 24, 15, and 36 by Listing Multiples

To calculate the LCM of 24, 15, 36 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 . . .), 15 (15, 30, 45, 60, 75 . . .), and 36 (36, 72, 108, 144, 180 . . .).
Step 2: The common multiples from the multiples of 24, 15, and 36 are 360, 720, . . .
Step 3: The smallest common multiple of 24, 15, and 36 is 360.

∴ The least common multiple of 24, 15, and 36 = 360.

☛ Also Check:

LCM of 24, 15, and 36 Examples

Example 1: Verify the relationship between the GCD and LCM of 24, 15, and 36.

Solution:

The relation between GCD and LCM of 24, 15, and 36 is given as,
LCM(24, 15, 36) = [(24 × 15 × 36) × GCD(24, 15, 36)]/[GCD(24, 15) × GCD(15, 36) × GCD(24, 36)]
⇒ Prime factorization of 24, 15 and 36:
- 24 = 2³ × 3¹
- 15 = 3¹ × 5¹
- 36 = 2² × 3²
∴ GCD of (24, 15), (15, 36), (24, 36) and (24, 15, 36) = 3, 3, 12 and 3 respectively.
Now, LHS = LCM(24, 15, 36) = 360.
And, RHS = [(24 × 15 × 36) × GCD(24, 15, 36)]/[GCD(24, 15) × GCD(15, 36) × GCD(24, 36)] = [(12960) × 3]/[3 × 3 × 12] = 360
LHS = RHS = 360.
Hence verified.
Example 2: Find the smallest number that is divisible by 24, 15, 36 exactly.

Solution:

The value of LCM(24, 15, 36) will be the smallest number that is exactly divisible by 24, 15, and 36.
⇒ Multiples of 24, 15, and 36:
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, . . . ., 312, 336, 360, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, . . . ., 330, 345, 360, . . . .
- Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, . . . ., 252, 288, 324, 360, . . . .
Therefore, the LCM of 24, 15, and 36 is 360.
Example 3: Calculate the LCM of 24, 15, and 36 using the GCD of the given numbers.

Solution:

Prime factorization of 24, 15, 36:
- 24 = 2³ × 3¹
- 15 = 3¹ × 5¹
- 36 = 2² × 3²
Therefore, GCD(24, 15) = 3, GCD(15, 36) = 3, GCD(24, 36) = 12, GCD(24, 15, 36) = 3
We know,
LCM(24, 15, 36) = [(24 × 15 × 36) × GCD(24, 15, 36)]/[GCD(24, 15) × GCD(15, 36) × GCD(24, 36)]
LCM(24, 15, 36) = (12960 × 3)/(3 × 3 × 12) = 360
⇒LCM(24, 15, 36) = 360

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FAQs on LCM of 24, 15, and 36

What is the LCM of 24, 15, and 36?

The LCM of 24, 15, and 36 is 360. To find the least common multiple (LCM) of 24, 15, and 36, we need to find the multiples of 24, 15, and 36 (multiples of 24 = 24, 48, 72, 96 . . . . 360 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 360 . . . . ; multiples of 36 = 36, 72, 108, 144 . . . . 360 . . . . ) and choose the smallest multiple that is exactly divisible by 24, 15, and 36, i.e., 360.

How to Find the LCM of 24, 15, and 36 by Prime Factorization?

To find the LCM of 24, 15, and 36 using prime factorization, we will find the prime factors, (24 = 2³ × 3¹), (15 = 3¹ × 5¹), and (36 = 2² × 3²). LCM of 24, 15, and 36 is the product of prime factors raised to their respective highest exponent among the numbers 24, 15, and 36.
⇒ LCM of 24, 15, 36 = 2³ × 3² × 5¹ = 360.

What is the Least Perfect Square Divisible by 24, 15, and 36?

The least number divisible by 24, 15, and 36 = LCM(24, 15, 36)
LCM of 24, 15, and 36 = 2 × 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 2, 5]
⇒ Least perfect square divisible by each 24, 15, and 36 = LCM(24, 15, 36) × 2 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.

What is the Relation Between GCF and LCM of 24, 15, 36?

The following equation can be used to express the relation between GCF and LCM of 24, 15, 36, i.e. LCM(24, 15, 36) = [(24 × 15 × 36) × GCF(24, 15, 36)]/[GCF(24, 15) × GCF(15, 36) × GCF(24, 36)].

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