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LCM of 96 and 108

LCM of 96 and 108 is the smallest number among all common multiples of 96 and 108. The first few multiples of 96 and 108 are (96, 192, 288, 384, 480, 576, . . . ) and (108, 216, 324, 432, 540, . . . ) respectively. There are 3 commonly used methods to find LCM of 96 and 108 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 96 and 108
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 96 and 108?

Answer: LCM of 96 and 108 is 864.

Explanation:

The LCM of two non-zero integers, x(96) and y(108), is the smallest positive integer m(864) that is divisible by both x(96) and y(108) without any remainder.

Methods to Find LCM of 96 and 108

The methods to find the LCM of 96 and 108 are explained below.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 96 and 108 by Prime Factorization

Prime factorization of 96 and 108 is (2 × 2 × 2 × 2 × 2 × 3) = 2⁵ × 3¹ and (2 × 2 × 3 × 3 × 3) = 2² × 3³ respectively. LCM of 96 and 108 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁵ × 3³ = 864.
Hence, the LCM of 96 and 108 by prime factorization is 864.

LCM of 96 and 108 by Listing Multiples

To calculate the LCM of 96 and 108 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 96 (96, 192, 288, 384, 480, 576, . . . ) and 108 (108, 216, 324, 432, 540, . . . . )
Step 2: The common multiples from the multiples of 96 and 108 are 864, 1728, . . .
Step 3: The smallest common multiple of 96 and 108 is 864.

∴ The least common multiple of 96 and 108 = 864.

LCM of 96 and 108 by Division Method

To calculate the LCM of 96 and 108 by the division method, we will divide the numbers(96, 108) by their prime factors (preferably common). The product of these divisors gives the LCM of 96 and 108.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 96 and 108. Write this prime number(2) on the left of the given numbers(96 and 108), separated as per the ladder arrangement.
Step 2: If any of the given numbers (96, 108) 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 96 and 108 is the product of all prime numbers on the left, i.e. LCM(96, 108) by division method = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 864.

☛ Also Check:

LCM of 96 and 108 Examples

Example 1: Find the smallest number that is divisible by 96 and 108 exactly.

Solution:

The smallest number that is divisible by 96 and 108 exactly is their LCM.
⇒ Multiples of 96 and 108:
- Multiples of 96 = 96, 192, 288, 384, 480, 576, 672, 768, 864, . . . .
- Multiples of 108 = 108, 216, 324, 432, 540, 648, 756, 864, . . . .
Therefore, the LCM of 96 and 108 is 864.
Example 2: The GCD and LCM of two numbers are 12 and 864 respectively. If one number is 108, find the other number.

Solution:

Let the other number be m.
∵ GCD × LCM = 108 × m
⇒ m = (GCD × LCM)/108
⇒ m = (12 × 864)/108
⇒ m = 96
Therefore, the other number is 96.
Example 3: Verify the relationship between GCF and LCM of 96 and 108.

Solution:

The relation between GCF and LCM of 96 and 108 is given as,
LCM(96, 108) × GCF(96, 108) = Product of 96, 108
Prime factorization of 96 and 108 is given as, 96 = (2 × 2 × 2 × 2 × 2 × 3) = 2⁵ × 3¹ and 108 = (2 × 2 × 3 × 3 × 3) = 2² × 3³
LCM(96, 108) = 864
GCF(96, 108) = 12
LHS = LCM(96, 108) × GCF(96, 108) = 864 × 12 = 10368
RHS = Product of 96, 108 = 96 × 108 = 10368
⇒ LHS = RHS = 10368
Hence, verified.

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FAQs on LCM of 96 and 108

What is the LCM of 96 and 108?

The LCM of 96 and 108 is 864. To find the least common multiple of 96 and 108, we need to find the multiples of 96 and 108 (multiples of 96 = 96, 192, 288, 384 . . . . 864; multiples of 108 = 108, 216, 324, 432 . . . . 864) and choose the smallest multiple that is exactly divisible by 96 and 108, i.e., 864.

What is the Relation Between GCF and LCM of 96, 108?

The following equation can be used to express the relation between GCF and LCM of 96 and 108, i.e. GCF × LCM = 96 × 108.

If the LCM of 108 and 96 is 864, Find its GCF.

LCM(108, 96) × GCF(108, 96) = 108 × 96
Since the LCM of 108 and 96 = 864
⇒ 864 × GCF(108, 96) = 10368
Therefore, the GCF (greatest common factor) = 10368/864 = 12.

What is the Least Perfect Square Divisible by 96 and 108?

The least number divisible by 96 and 108 = LCM(96, 108)
LCM of 96 and 108 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 96 and 108 = LCM(96, 108) × 2 × 3 = 5184 [Square root of 5184 = √5184 = ±72]
Therefore, 5184 is the required number.

Which of the following is the LCM of 96 and 108? 45, 24, 864, 21

The value of LCM of 96, 108 is the smallest common multiple of 96 and 108. The number satisfying the given condition is 864.

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