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LCM of 16, 18, and 24

LCM of 16, 18, and 24 is the smallest number among all common multiples of 16, 18, and 24. The first few multiples of 16, 18, and 24 are (16, 32, 48, 64, 80 . . .), (18, 36, 54, 72, 90 . . .), and (24, 48, 72, 96, 120 . . .) respectively. There are 3 commonly used methods to find LCM of 16, 18, 24 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 16, 18, and 24
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 16, 18, and 24?

Answer: LCM of 16, 18, and 24 is 144.

Explanation:

The LCM of three non-zero integers, a(16), b(18), and c(24), is the smallest positive integer m(144) that is divisible by a(16), b(18), and c(24) without any remainder.

Methods to Find LCM of 16, 18, and 24

The methods to find the LCM of 16, 18, and 24 are explained below.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 16, 18, and 24 by Division Method

To calculate the LCM of 16, 18, and 24 by the division method, we will divide the numbers(16, 18, 24) by their prime factors (preferably common). The product of these divisors gives the LCM of 16, 18, and 24.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 16, 18, and 24. Write this prime number(2) on the left of the given numbers(16, 18, and 24), separated as per the ladder arrangement.
Step 2: If any of the given numbers (16, 18, 24) 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, 18, and 24 is the product of all prime numbers on the left, i.e. LCM(16, 18, 24) by division method = 2 × 2 × 2 × 2 × 3 × 3 = 144.

LCM of 16, 18, and 24 by Listing Multiples

To calculate the LCM of 16, 18, 24 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 . . .), 18 (18, 36, 54, 72, 90 . . .), and 24 (24, 48, 72, 96, 120 . . .).
Step 2: The common multiples from the multiples of 16, 18, and 24 are 144, 288, . . .
Step 3: The smallest common multiple of 16, 18, and 24 is 144.

∴ The least common multiple of 16, 18, and 24 = 144.

LCM of 16, 18, and 24 by Prime Factorization

Prime factorization of 16, 18, and 24 is (2 × 2 × 2 × 2) = 2⁴, (2 × 3 × 3) = 2¹ × 3², and (2 × 2 × 2 × 3) = 2³ × 3¹ respectively. LCM of 16, 18, and 24 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3² = 144.
Hence, the LCM of 16, 18, and 24 by prime factorization is 144.

☛ Also Check:

LCM of 16, 18, and 24 Examples

Example 1: Verify the relationship between the GCD and LCM of 16, 18, and 24.

Solution:

The relation between GCD and LCM of 16, 18, and 24 is given as,
LCM(16, 18, 24) = [(16 × 18 × 24) × GCD(16, 18, 24)]/[GCD(16, 18) × GCD(18, 24) × GCD(16, 24)]
⇒ Prime factorization of 16, 18 and 24:
- 16 = 2⁴
- 18 = 2¹ × 3²
- 24 = 2³ × 3¹
∴ GCD of (16, 18), (18, 24), (16, 24) and (16, 18, 24) = 2, 6, 8 and 2 respectively.
Now, LHS = LCM(16, 18, 24) = 144.
And, RHS = [(16 × 18 × 24) × GCD(16, 18, 24)]/[GCD(16, 18) × GCD(18, 24) × GCD(16, 24)] = [(6912) × 2]/[2 × 6 × 8] = 144
LHS = RHS = 144.
Hence verified.
Example 2: Find the smallest number that is divisible by 16, 18, 24 exactly.

Solution:

The smallest number that is divisible by 16, 18, and 24 exactly is their LCM.
⇒ Multiples of 16, 18, and 24:
- Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, . . . .
- Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, 144, . . . .
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, . . . .
Therefore, the LCM of 16, 18, and 24 is 144.
Example 3: Calculate the LCM of 16, 18, and 24 using the GCD of the given numbers.

Solution:

Prime factorization of 16, 18, 24:
- 16 = 2⁴
- 18 = 2¹ × 3²
- 24 = 2³ × 3¹
Therefore, GCD(16, 18) = 2, GCD(18, 24) = 6, GCD(16, 24) = 8, GCD(16, 18, 24) = 2
We know,
LCM(16, 18, 24) = [(16 × 18 × 24) × GCD(16, 18, 24)]/[GCD(16, 18) × GCD(18, 24) × GCD(16, 24)]
LCM(16, 18, 24) = (6912 × 2)/(2 × 6 × 8) = 144
⇒LCM(16, 18, 24) = 144

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FAQs on LCM of 16, 18, and 24

What is the LCM of 16, 18, and 24?

The LCM of 16, 18, and 24 is 144. To find the LCM of 16, 18, and 24, we need to find the multiples of 16, 18, and 24 (multiples of 16 = 16, 32, 48, 64 . . . . 144 . . . . ; multiples of 18 = 18, 36, 54, 72 . . . . 144 . . . . ; multiples of 24 = 24, 48, 72, 96, 144 . . . .) and choose the smallest multiple that is exactly divisible by 16, 18, and 24, i.e., 144.

What is the Relation Between GCF and LCM of 16, 18, 24?

The following equation can be used to express the relation between GCF and LCM of 16, 18, 24, i.e. LCM(16, 18, 24) = [(16 × 18 × 24) × GCF(16, 18, 24)]/[GCF(16, 18) × GCF(18, 24) × GCF(16, 24)].

What are the Methods to Find LCM of 16, 18, 24?

The commonly used methods to find the LCM of 16, 18, 24 are:

Division Method
Listing Multiples
Prime Factorization Method

What is the Least Perfect Square Divisible by 16, 18, and 24?

The least number divisible by 16, 18, and 24 = LCM(16, 18, 24)
LCM of 16, 18, and 24 = 2 × 2 × 2 × 2 × 3 × 3 [No incomplete pair]
⇒ Least perfect square divisible by each 16, 18, and 24 = LCM(16, 18, 24) = 144 [Square root of 144 = √144 = ±12]
Therefore, 144 is the required number.

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