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LCM of 48 and 108

LCM of 48 and 108 is the smallest number among all common multiples of 48 and 108. The first few multiples of 48 and 108 are (48, 96, 144, 192, 240, 288, . . . ) and (108, 216, 324, 432, . . . ) respectively. There are 3 commonly used methods to find LCM of 48 and 108 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 48 and 108
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 48 and 108?

Answer: LCM of 48 and 108 is 432.

Explanation:

The LCM of two non-zero integers, x(48) and y(108), is the smallest positive integer m(432) that is divisible by both x(48) and y(108) without any remainder.

Methods to Find LCM of 48 and 108

Let's look at the different methods for finding the LCM of 48 and 108.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 48 and 108 by Division Method

To calculate the LCM of 48 and 108 by the division method, we will divide the numbers(48, 108) by their prime factors (preferably common). The product of these divisors gives the LCM of 48 and 108.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 48 and 108. Write this prime number(2) on the left of the given numbers(48 and 108), separated as per the ladder arrangement.
Step 2: If any of the given numbers (48, 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 48 and 108 is the product of all prime numbers on the left, i.e. LCM(48, 108) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 432.

LCM of 48 and 108 by Listing Multiples

To calculate the LCM of 48 and 108 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 48 (48, 96, 144, 192, 240, 288, . . . ) and 108 (108, 216, 324, 432, . . . . )
Step 2: The common multiples from the multiples of 48 and 108 are 432, 864, . . .
Step 3: The smallest common multiple of 48 and 108 is 432.

∴ The least common multiple of 48 and 108 = 432.

LCM of 48 and 108 by Prime Factorization

Prime factorization of 48 and 108 is (2 × 2 × 2 × 2 × 3) = 2⁴ × 3¹ and (2 × 2 × 3 × 3 × 3) = 2² × 3³ respectively. LCM of 48 and 108 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3³ = 432.
Hence, the LCM of 48 and 108 by prime factorization is 432.

☛ Also Check:

LCM of 48 and 108 Examples

Example 1: The GCD and LCM of two numbers are 12 and 432 respectively. If one number is 48, find the other number.

Solution:

Let the other number be y.
∵ GCD × LCM = 48 × y
⇒ y = (GCD × LCM)/48
⇒ y = (12 × 432)/48
⇒ y = 108
Therefore, the other number is 108.
Example 2: Verify the relationship between GCF and LCM of 48 and 108.

Solution:

The relation between GCF and LCM of 48 and 108 is given as,
LCM(48, 108) × GCF(48, 108) = Product of 48, 108
Prime factorization of 48 and 108 is given as, 48 = (2 × 2 × 2 × 2 × 3) = 2⁴ × 3¹ and 108 = (2 × 2 × 3 × 3 × 3) = 2² × 3³
LCM(48, 108) = 432
GCF(48, 108) = 12
LHS = LCM(48, 108) × GCF(48, 108) = 432 × 12 = 5184
RHS = Product of 48, 108 = 48 × 108 = 5184
⇒ LHS = RHS = 5184
Hence, verified.
Example 3: Find the smallest number that is divisible by 48 and 108 exactly.

Solution:

The smallest number that is divisible by 48 and 108 exactly is their LCM.
⇒ Multiples of 48 and 108:
- Multiples of 48 = 48, 96, 144, 192, 240, 288, 336, 384, 432, . . . .
- Multiples of 108 = 108, 216, 324, 432, 540, 648, . . . .
Therefore, the LCM of 48 and 108 is 432.

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FAQs on LCM of 48 and 108

What is the LCM of 48 and 108?

The LCM of 48 and 108 is 432. To find the least common multiple of 48 and 108, we need to find the multiples of 48 and 108 (multiples of 48 = 48, 96, 144, 192 . . . . 432; multiples of 108 = 108, 216, 324, 432) and choose the smallest multiple that is exactly divisible by 48 and 108, i.e., 432.

How to Find the LCM of 48 and 108 by Prime Factorization?

To find the LCM of 48 and 108 using prime factorization, we will find the prime factors, (48 = 2 × 2 × 2 × 2 × 3) and (108 = 2 × 2 × 3 × 3 × 3). LCM of 48 and 108 is the product of prime factors raised to their respective highest exponent among the numbers 48 and 108.
⇒ LCM of 48, 108 = 2⁴ × 3³ = 432.

What is the Relation Between GCF and LCM of 48, 108?

The following equation can be used to express the relation between GCF and LCM of 48 and 108, i.e. GCF × LCM = 48 × 108.

If the LCM of 108 and 48 is 432, Find its GCF.

LCM(108, 48) × GCF(108, 48) = 108 × 48
Since the LCM of 108 and 48 = 432
⇒ 432 × GCF(108, 48) = 5184
Therefore, the greatest common factor (GCF) = 5184/432 = 12.

What is the Least Perfect Square Divisible by 48 and 108?

The least number divisible by 48 and 108 = LCM(48, 108)
LCM of 48 and 108 = 2 × 2 × 2 × 2 × 3 × 3 × 3 [Incomplete pair(s): 3]
⇒ Least perfect square divisible by each 48 and 108 = LCM(48, 108) × 3 = 1296 [Square root of 1296 = √1296 = ±36]
Therefore, 1296 is the required number.

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