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LCM of 24, 30, and 40

LCM of 24, 30, and 40 is the smallest number among all common multiples of 24, 30, and 40. The first few multiples of 24, 30, and 40 are (24, 48, 72, 96, 120 . . .), (30, 60, 90, 120, 150 . . .), and (40, 80, 120, 160, 200 . . .) respectively. There are 3 commonly used methods to find LCM of 24, 30, 40 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 24, 30, and 40
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 24, 30, and 40?

Answer: LCM of 24, 30, and 40 is 120.

Explanation:

The LCM of three non-zero integers, a(24), b(30), and c(40), is the smallest positive integer m(120) that is divisible by a(24), b(30), and c(40) without any remainder.

Methods to Find LCM of 24, 30, and 40

Let's look at the different methods for finding the LCM of 24, 30, and 40.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 24, 30, and 40 by Prime Factorization

Prime factorization of 24, 30, and 40 is (2 × 2 × 2 × 3) = 2³ × 3¹, (2 × 3 × 5) = 2¹ × 3¹ × 5¹, and (2 × 2 × 2 × 5) = 2³ × 5¹ respectively. LCM of 24, 30, and 40 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3¹ × 5¹ = 120.
Hence, the LCM of 24, 30, and 40 by prime factorization is 120.

LCM of 24, 30, and 40 by Listing Multiples

To calculate the LCM of 24, 30, 40 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 . . .), 30 (30, 60, 90, 120, 150 . . .), and 40 (40, 80, 120, 160, 200 . . .).
Step 2: The common multiples from the multiples of 24, 30, and 40 are 120, 240, . . .
Step 3: The smallest common multiple of 24, 30, and 40 is 120.

∴ The least common multiple of 24, 30, and 40 = 120.

LCM of 24, 30, and 40 by Division Method

To calculate the LCM of 24, 30, and 40 by the division method, we will divide the numbers(24, 30, 40) by their prime factors (preferably common). The product of these divisors gives the LCM of 24, 30, and 40.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 24, 30, and 40. Write this prime number(2) on the left of the given numbers(24, 30, and 40), separated as per the ladder arrangement.
Step 2: If any of the given numbers (24, 30, 40) 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, 30, and 40 is the product of all prime numbers on the left, i.e. LCM(24, 30, 40) by division method = 2 × 2 × 2 × 3 × 5 = 120.

☛ Also Check:

LCM of 24, 30, and 40 Examples

Example 1: Verify the relationship between the GCD and LCM of 24, 30, and 40.

Solution:

The relation between GCD and LCM of 24, 30, and 40 is given as,
LCM(24, 30, 40) = [(24 × 30 × 40) × GCD(24, 30, 40)]/[GCD(24, 30) × GCD(30, 40) × GCD(24, 40)]
⇒ Prime factorization of 24, 30 and 40:
- 24 = 2³ × 3¹
- 30 = 2¹ × 3¹ × 5¹
- 40 = 2³ × 5¹
∴ GCD of (24, 30), (30, 40), (24, 40) and (24, 30, 40) = 6, 10, 8 and 2 respectively.
Now, LHS = LCM(24, 30, 40) = 120.
And, RHS = [(24 × 30 × 40) × GCD(24, 30, 40)]/[GCD(24, 30) × GCD(30, 40) × GCD(24, 40)] = [(28800) × 2]/[6 × 10 × 8] = 120
LHS = RHS = 120.
Hence verified.
Example 2: Calculate the LCM of 24, 30, and 40 using the GCD of the given numbers.

Solution:

Prime factorization of 24, 30, 40:
- 24 = 2³ × 3¹
- 30 = 2¹ × 3¹ × 5¹
- 40 = 2³ × 5¹
Therefore, GCD(24, 30) = 6, GCD(30, 40) = 10, GCD(24, 40) = 8, GCD(24, 30, 40) = 2
We know,
LCM(24, 30, 40) = [(24 × 30 × 40) × GCD(24, 30, 40)]/[GCD(24, 30) × GCD(30, 40) × GCD(24, 40)]
LCM(24, 30, 40) = (28800 × 2)/(6 × 10 × 8) = 120
⇒LCM(24, 30, 40) = 120
Example 3: Find the smallest number that is divisible by 24, 30, 40 exactly.

Solution:

The smallest number that is divisible by 24, 30, and 40 exactly is their LCM.
⇒ Multiples of 24, 30, and 40:
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, . . . .
- Multiples of 30 = 30, 60, 90, 120, 150, 180, 210, . . . .
- Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, . . . .
Therefore, the LCM of 24, 30, and 40 is 120.

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FAQs on LCM of 24, 30, and 40

What is the LCM of 24, 30, and 40?

The LCM of 24, 30, and 40 is 120. To find the LCM (least common multiple) of 24, 30, and 40, we need to find the multiples of 24, 30, and 40 (multiples of 24 = 24, 48, 72, 96 . . . .; multiples of 30 = 30, 60, 90, 120 . . . .; multiples of 40 = 40, 80, 120, 160 . . . .) and choose the smallest multiple that is exactly divisible by 24, 30, and 40, i.e., 120.

What is the Least Perfect Square Divisible by 24, 30, and 40?

The least number divisible by 24, 30, and 40 = LCM(24, 30, 40)
LCM of 24, 30, and 40 = 2 × 2 × 2 × 3 × 5 [Incomplete pair(s): 2, 3, 5]
⇒ Least perfect square divisible by each 24, 30, and 40 = LCM(24, 30, 40) × 2 × 3 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.

How to Find the LCM of 24, 30, and 40 by Prime Factorization?

To find the LCM of 24, 30, and 40 using prime factorization, we will find the prime factors, (24 = 2³ × 3¹), (30 = 2¹ × 3¹ × 5¹), and (40 = 2³ × 5¹). LCM of 24, 30, and 40 is the product of prime factors raised to their respective highest exponent among the numbers 24, 30, and 40.
⇒ LCM of 24, 30, 40 = 2³ × 3¹ × 5¹ = 120.

What is the Relation Between GCF and LCM of 24, 30, 40?

The following equation can be used to express the relation between GCF and LCM of 24, 30, 40, i.e. LCM(24, 30, 40) = [(24 × 30 × 40) × GCF(24, 30, 40)]/[GCF(24, 30) × GCF(30, 40) × GCF(24, 40)].

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