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LCM of 48 and 64

LCM of 48 and 64 is the smallest number among all common multiples of 48 and 64. The first few multiples of 48 and 64 are (48, 96, 144, 192, 240, 288, . . . ) and (64, 128, 192, 256, . . . ) respectively. There are 3 commonly used methods to find LCM of 48 and 64 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 48 and 64
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 48 and 64?

Answer: LCM of 48 and 64 is 192.

Explanation:

The LCM of two non-zero integers, x(48) and y(64), is the smallest positive integer m(192) that is divisible by both x(48) and y(64) without any remainder.

Methods to Find LCM of 48 and 64

Let's look at the different methods for finding the LCM of 48 and 64.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 48 and 64 by Prime Factorization

Prime factorization of 48 and 64 is (2 × 2 × 2 × 2 × 3) = 2⁴ × 3¹ and (2 × 2 × 2 × 2 × 2 × 2) = 2⁶ respectively. LCM of 48 and 64 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁶ × 3¹ = 192.
Hence, the LCM of 48 and 64 by prime factorization is 192.

LCM of 48 and 64 by Listing Multiples

To calculate the LCM of 48 and 64 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 64 (64, 128, 192, 256, . . . . )
Step 2: The common multiples from the multiples of 48 and 64 are 192, 384, . . .
Step 3: The smallest common multiple of 48 and 64 is 192.

∴ The least common multiple of 48 and 64 = 192.

LCM of 48 and 64 by Division Method

To calculate the LCM of 48 and 64 by the division method, we will divide the numbers(48, 64) by their prime factors (preferably common). The product of these divisors gives the LCM of 48 and 64.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 48 and 64. Write this prime number(2) on the left of the given numbers(48 and 64), separated as per the ladder arrangement.
Step 2: If any of the given numbers (48, 64) 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 64 is the product of all prime numbers on the left, i.e. LCM(48, 64) by division method = 2 × 2 × 2 × 2 × 2 × 2 × 3 = 192.

☛ Also Check:

LCM of 48 and 64 Examples

Example 1: The product of two numbers is 3072. If their GCD is 16, what is their LCM?

Solution:

Given: GCD = 16
product of numbers = 3072
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 3072/16
Therefore, the LCM is 192.
The probable combination for the given case is LCM(48, 64) = 192.
Example 2: Find the smallest number that is divisible by 48 and 64 exactly.

Solution:

The smallest number that is divisible by 48 and 64 exactly is their LCM.
⇒ Multiples of 48 and 64:
- Multiples of 48 = 48, 96, 144, 192, 240, 288, . . . .
- Multiples of 64 = 64, 128, 192, 256, 320, 384, . . . .
Therefore, the LCM of 48 and 64 is 192.
Example 3: Verify the relationship between GCF and LCM of 48 and 64.

Solution:

The relation between GCF and LCM of 48 and 64 is given as,
LCM(48, 64) × GCF(48, 64) = Product of 48, 64
Prime factorization of 48 and 64 is given as, 48 = (2 × 2 × 2 × 2 × 3) = 2⁴ × 3¹ and 64 = (2 × 2 × 2 × 2 × 2 × 2) = 2⁶
LCM(48, 64) = 192
GCF(48, 64) = 16
LHS = LCM(48, 64) × GCF(48, 64) = 192 × 16 = 3072
RHS = Product of 48, 64 = 48 × 64 = 3072
⇒ LHS = RHS = 3072
Hence, verified.

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FAQs on LCM of 48 and 64

What is the LCM of 48 and 64?

The LCM of 48 and 64 is 192. To find the least common multiple (LCM) of 48 and 64, we need to find the multiples of 48 and 64 (multiples of 48 = 48, 96, 144, 192; multiples of 64 = 64, 128, 192, 256) and choose the smallest multiple that is exactly divisible by 48 and 64, i.e., 192.

Which of the following is the LCM of 48 and 64? 192, 40, 20, 35

The value of LCM of 48, 64 is the smallest common multiple of 48 and 64. The number satisfying the given condition is 192.

If the LCM of 64 and 48 is 192, Find its GCF.

LCM(64, 48) × GCF(64, 48) = 64 × 48
Since the LCM of 64 and 48 = 192
⇒ 192 × GCF(64, 48) = 3072
Therefore, the greatest common factor = 3072/192 = 16.

What are the Methods to Find LCM of 48 and 64?

The commonly used methods to find the LCM of 48 and 64 are:

Listing Multiples
Prime Factorization Method
Division Method

What is the Relation Between GCF and LCM of 48, 64?

The following equation can be used to express the relation between GCF and LCM of 48 and 64, i.e. GCF × LCM = 48 × 64.

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