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LCM of 16, 24, and 40

LCM of 16, 24, and 40 is the smallest number among all common multiples of 16, 24, and 40. The first few multiples of 16, 24, and 40 are (16, 32, 48, 64, 80 . . .), (24, 48, 72, 96, 120 . . .), and (40, 80, 120, 160, 200 . . .) respectively. There are 3 commonly used methods to find LCM of 16, 24, 40 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 16, 24, and 40
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 16, 24, and 40?

Answer: LCM of 16, 24, and 40 is 240.

Explanation:

The LCM of three non-zero integers, a(16), b(24), and c(40), is the smallest positive integer m(240) that is divisible by a(16), b(24), and c(40) without any remainder.

Methods to Find LCM of 16, 24, and 40

The methods to find the LCM of 16, 24, and 40 are explained below.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 16, 24, and 40 by Division Method

To calculate the LCM of 16, 24, and 40 by the division method, we will divide the numbers(16, 24, 40) by their prime factors (preferably common). The product of these divisors gives the LCM of 16, 24, and 40.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 16, 24, and 40. Write this prime number(2) on the left of the given numbers(16, 24, and 40), separated as per the ladder arrangement.
Step 2: If any of the given numbers (16, 24, 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 16, 24, and 40 is the product of all prime numbers on the left, i.e. LCM(16, 24, 40) by division method = 2 × 2 × 2 × 2 × 3 × 5 = 240.

LCM of 16, 24, and 40 by Prime Factorization

Prime factorization of 16, 24, and 40 is (2 × 2 × 2 × 2) = 2⁴, (2 × 2 × 2 × 3) = 2³ × 3¹, and (2 × 2 × 2 × 5) = 2³ × 5¹ respectively. LCM of 16, 24, and 40 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3¹ × 5¹ = 240.
Hence, the LCM of 16, 24, and 40 by prime factorization is 240.

LCM of 16, 24, and 40 by Listing Multiples

To calculate the LCM of 16, 24, 40 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 16 (16, 32, 48, 64, 80 . . .), 24 (24, 48, 72, 96, 120 . . .), and 40 (40, 80, 120, 160, 200 . . .).
Step 2: The common multiples from the multiples of 16, 24, and 40 are 240, 480, . . .
Step 3: The smallest common multiple of 16, 24, and 40 is 240.

∴ The least common multiple of 16, 24, and 40 = 240.

☛ Also Check:

LCM of 16, 24, and 40 Examples

Example 1: Find the smallest number that is divisible by 16, 24, 40 exactly.

Solution:

The smallest number that is divisible by 16, 24, and 40 exactly is their LCM.
⇒ Multiples of 16, 24, and 40:
- Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, . . . .
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, . . . .
- Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, . . . .
Therefore, the LCM of 16, 24, and 40 is 240.
Example 2: Calculate the LCM of 16, 24, and 40 using the GCD of the given numbers.

Solution:

Prime factorization of 16, 24, 40:
- 16 = 2⁴
- 24 = 2³ × 3¹
- 40 = 2³ × 5¹
Therefore, GCD(16, 24) = 8, GCD(24, 40) = 8, GCD(16, 40) = 8, GCD(16, 24, 40) = 8
We know,
LCM(16, 24, 40) = [(16 × 24 × 40) × GCD(16, 24, 40)]/[GCD(16, 24) × GCD(24, 40) × GCD(16, 40)]
LCM(16, 24, 40) = (15360 × 8)/(8 × 8 × 8) = 240
⇒LCM(16, 24, 40) = 240
Example 3: Verify the relationship between the GCD and LCM of 16, 24, and 40.

Solution:

The relation between GCD and LCM of 16, 24, and 40 is given as,
LCM(16, 24, 40) = [(16 × 24 × 40) × GCD(16, 24, 40)]/[GCD(16, 24) × GCD(24, 40) × GCD(16, 40)]
⇒ Prime factorization of 16, 24 and 40:
- 16 = 2⁴
- 24 = 2³ × 3¹
- 40 = 2³ × 5¹
∴ GCD of (16, 24), (24, 40), (16, 40) and (16, 24, 40) = 8, 8, 8 and 8 respectively.
Now, LHS = LCM(16, 24, 40) = 240.
And, RHS = [(16 × 24 × 40) × GCD(16, 24, 40)]/[GCD(16, 24) × GCD(24, 40) × GCD(16, 40)] = [(15360) × 8]/[8 × 8 × 8] = 240
LHS = RHS = 240.
Hence verified.

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FAQs on LCM of 16, 24, and 40

What is the LCM of 16, 24, and 40?

The LCM of 16, 24, and 40 is 240. To find the least common multiple (LCM) of 16, 24, and 40, we need to find the multiples of 16, 24, and 40 (multiples of 16 = 16, 32, 48, 64 . . . . 240 . . . . ; multiples of 24 = 24, 48, 72, 96 . . . . 240 . . . . ; multiples of 40 = 40, 80, 120, 160, 240 . . . .) and choose the smallest multiple that is exactly divisible by 16, 24, and 40, i.e., 240.

What are the Methods to Find LCM of 16, 24, 40?

The commonly used methods to find the LCM of 16, 24, 40 are:

Prime Factorization Method
Division Method
Listing Multiples

What is the Least Perfect Square Divisible by 16, 24, and 40?

The least number divisible by 16, 24, and 40 = LCM(16, 24, 40)
LCM of 16, 24, and 40 = 2 × 2 × 2 × 2 × 3 × 5 [Incomplete pair(s): 3, 5]
⇒ Least perfect square divisible by each 16, 24, and 40 = LCM(16, 24, 40) × 3 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.

What is the Relation Between GCF and LCM of 16, 24, 40?

The following equation can be used to express the relation between GCF and LCM of 16, 24, 40, i.e. LCM(16, 24, 40) = [(16 × 24 × 40) × GCF(16, 24, 40)]/[GCF(16, 24) × GCF(24, 40) × GCF(16, 40)].

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