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LCM of 18, 24, and 32

LCM of 18, 24, and 32 is the smallest number among all common multiples of 18, 24, and 32. The first few multiples of 18, 24, and 32 are (18, 36, 54, 72, 90 . . .), (24, 48, 72, 96, 120 . . .), and (32, 64, 96, 128, 160 . . .) respectively. There are 3 commonly used methods to find LCM of 18, 24, 32 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 18, 24, and 32
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 18, 24, and 32?

Answer: LCM of 18, 24, and 32 is 288.

Explanation:

The LCM of three non-zero integers, a(18), b(24), and c(32), is the smallest positive integer m(288) that is divisible by a(18), b(24), and c(32) without any remainder.

Methods to Find LCM of 18, 24, and 32

Let's look at the different methods for finding the LCM of 18, 24, and 32.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 18, 24, and 32 by Prime Factorization

Prime factorization of 18, 24, and 32 is (2 × 3 × 3) = 2¹ × 3², (2 × 2 × 2 × 3) = 2³ × 3¹, and (2 × 2 × 2 × 2 × 2) = 2⁵ respectively. LCM of 18, 24, and 32 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁵ × 3² = 288.
Hence, the LCM of 18, 24, and 32 by prime factorization is 288.

LCM of 18, 24, and 32 by Listing Multiples

To calculate the LCM of 18, 24, 32 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 18 (18, 36, 54, 72, 90 . . .), 24 (24, 48, 72, 96, 120 . . .), and 32 (32, 64, 96, 128, 160 . . .).
Step 2: The common multiples from the multiples of 18, 24, and 32 are 288, 576, . . .
Step 3: The smallest common multiple of 18, 24, and 32 is 288.

∴ The least common multiple of 18, 24, and 32 = 288.

LCM of 18, 24, and 32 by Division Method

To calculate the LCM of 18, 24, and 32 by the division method, we will divide the numbers(18, 24, 32) by their prime factors (preferably common). The product of these divisors gives the LCM of 18, 24, and 32.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 18, 24, and 32. Write this prime number(2) on the left of the given numbers(18, 24, and 32), separated as per the ladder arrangement.
Step 2: If any of the given numbers (18, 24, 32) 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 18, 24, and 32 is the product of all prime numbers on the left, i.e. LCM(18, 24, 32) by division method = 2 × 2 × 2 × 2 × 2 × 3 × 3 = 288.

☛ Also Check:

LCM of 18, 24, and 32 Examples

Example 1: Verify the relationship between the GCD and LCM of 18, 24, and 32.

Solution:

The relation between GCD and LCM of 18, 24, and 32 is given as,
LCM(18, 24, 32) = [(18 × 24 × 32) × GCD(18, 24, 32)]/[GCD(18, 24) × GCD(24, 32) × GCD(18, 32)]
⇒ Prime factorization of 18, 24 and 32:
- 18 = 2¹ × 3²
- 24 = 2³ × 3¹
- 32 = 2⁵
∴ GCD of (18, 24), (24, 32), (18, 32) and (18, 24, 32) = 6, 8, 2 and 2 respectively.
Now, LHS = LCM(18, 24, 32) = 288.
And, RHS = [(18 × 24 × 32) × GCD(18, 24, 32)]/[GCD(18, 24) × GCD(24, 32) × GCD(18, 32)] = [(13824) × 2]/[6 × 8 × 2] = 288
LHS = RHS = 288.
Hence verified.
Example 2: Calculate the LCM of 18, 24, and 32 using the GCD of the given numbers.

Solution:

Prime factorization of 18, 24, 32:
- 18 = 2¹ × 3²
- 24 = 2³ × 3¹
- 32 = 2⁵
Therefore, GCD(18, 24) = 6, GCD(24, 32) = 8, GCD(18, 32) = 2, GCD(18, 24, 32) = 2
We know,
LCM(18, 24, 32) = [(18 × 24 × 32) × GCD(18, 24, 32)]/[GCD(18, 24) × GCD(24, 32) × GCD(18, 32)]
LCM(18, 24, 32) = (13824 × 2)/(6 × 8 × 2) = 288
⇒LCM(18, 24, 32) = 288
Example 3: Find the smallest number that is divisible by 18, 24, 32 exactly.

Solution:

The value of LCM(18, 24, 32) will be the smallest number that is exactly divisible by 18, 24, and 32.
⇒ Multiples of 18, 24, and 32:
- Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, . . . ., 234, 252, 270, 288, . . . .
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, . . . ., 192, 216, 240, 264, 288, . . . .
- Multiples of 32 = 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, . . . ., 160, 192, 224, 256, 288, . . . .
Therefore, the LCM of 18, 24, and 32 is 288.

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FAQs on LCM of 18, 24, and 32

What is the LCM of 18, 24, and 32?

The LCM of 18, 24, and 32 is 288. To find the least common multiple (LCM) of 18, 24, and 32, we need to find the multiples of 18, 24, and 32 (multiples of 18 = 18, 36, 54, 72 . . . . 288 . . . . ; multiples of 24 = 24, 48, 72, 96 . . . . 288 . . . . ; multiples of 32 = 32, 64, 96, 128 . . . . 288 . . . . ) and choose the smallest multiple that is exactly divisible by 18, 24, and 32, i.e., 288.

Which of the following is the LCM of 18, 24, and 32? 120, 110, 81, 288

The value of LCM of 18, 24, 32 is the smallest common multiple of 18, 24, and 32. The number satisfying the given condition is 288.

How to Find the LCM of 18, 24, and 32 by Prime Factorization?

To find the LCM of 18, 24, and 32 using prime factorization, we will find the prime factors, (18 = 2¹ × 3²), (24 = 2³ × 3¹), and (32 = 2⁵). LCM of 18, 24, and 32 is the product of prime factors raised to their respective highest exponent among the numbers 18, 24, and 32.
⇒ LCM of 18, 24, 32 = 2⁵ × 3² = 288.

What is the Least Perfect Square Divisible by 18, 24, and 32?

The least number divisible by 18, 24, and 32 = LCM(18, 24, 32)
LCM of 18, 24, and 32 = 2 × 2 × 2 × 2 × 2 × 3 × 3 [Incomplete pair(s): 2]
⇒ Least perfect square divisible by each 18, 24, and 32 = LCM(18, 24, 32) × 2 = 576 [Square root of 576 = √576 = ±24]
Therefore, 576 is the required number.

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