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LCM of 48 and 54

LCM of 48 and 54 is the smallest number among all common multiples of 48 and 54. The first few multiples of 48 and 54 are (48, 96, 144, 192, . . . ) and (54, 108, 162, 216, 270, 324, . . . ) respectively. There are 3 commonly used methods to find LCM of 48 and 54 - by listing multiples, by division method, and by prime factorization.

1. LCM of 48 and 54
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 48 and 54?

Answer: LCM of 48 and 54 is 432.

LCM of 48 and 54

Explanation:

The LCM of two non-zero integers, x(48) and y(54), is the smallest positive integer m(432) that is divisible by both x(48) and y(54) without any remainder.

Methods to Find LCM of 48 and 54

The methods to find the LCM of 48 and 54 are explained below.

  • By Division Method
  • By Listing Multiples
  • By Prime Factorization Method

LCM of 48 and 54 by Division Method

LCM of 48 and 54 by Division Method

To calculate the LCM of 48 and 54 by the division method, we will divide the numbers(48, 54) by their prime factors (preferably common). The product of these divisors gives the LCM of 48 and 54.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 48 and 54. Write this prime number(2) on the left of the given numbers(48 and 54), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (48, 54) 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 54 is the product of all prime numbers on the left, i.e. LCM(48, 54) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 432.

LCM of 48 and 54 by Listing Multiples

To calculate the LCM of 48 and 54 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, . . . ) and 54 (54, 108, 162, 216, 270, 324, . . . . )
  • Step 2: The common multiples from the multiples of 48 and 54 are 432, 864, . . .
  • Step 3: The smallest common multiple of 48 and 54 is 432.

∴ The least common multiple of 48 and 54 = 432.

LCM of 48 and 54 by Prime Factorization

Prime factorization of 48 and 54 is (2 × 2 × 2 × 2 × 3) = 24 × 31 and (2 × 3 × 3 × 3) = 21 × 33 respectively. LCM of 48 and 54 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 24 × 33 = 432.
Hence, the LCM of 48 and 54 by prime factorization is 432.

☛ Also Check:

LCM of 48 and 54 Examples

  1. Example 1: Verify the relationship between GCF and LCM of 48 and 54.

    Solution:

    The relation between GCF and LCM of 48 and 54 is given as,
    LCM(48, 54) × GCF(48, 54) = Product of 48, 54
    Prime factorization of 48 and 54 is given as, 48 = (2 × 2 × 2 × 2 × 3) = 24 × 31 and 54 = (2 × 3 × 3 × 3) = 21 × 33
    LCM(48, 54) = 432
    GCF(48, 54) = 6
    LHS = LCM(48, 54) × GCF(48, 54) = 432 × 6 = 2592
    RHS = Product of 48, 54 = 48 × 54 = 2592
    ⇒ LHS = RHS = 2592
    Hence, verified.

  2. Example 2: Find the smallest number that is divisible by 48 and 54 exactly.

    Solution:

    The smallest number that is divisible by 48 and 54 exactly is their LCM.
    ⇒ Multiples of 48 and 54:

    • Multiples of 48 = 48, 96, 144, 192, 240, 288, 336, 384, 432, . . . .
    • Multiples of 54 = 54, 108, 162, 216, 270, 324, 378, 432, . . . .

    Therefore, the LCM of 48 and 54 is 432.

  3. Example 3: The GCD and LCM of two numbers are 6 and 432 respectively. If one number is 48, find the other number.

    Solution:

    Let the other number be a.
    ∵ GCD × LCM = 48 × a
    ⇒ a = (GCD × LCM)/48
    ⇒ a = (6 × 432)/48
    ⇒ a = 54
    Therefore, the other number is 54.

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FAQs on LCM of 48 and 54

What is the LCM of 48 and 54?

The LCM of 48 and 54 is 432. To find the LCM (least common multiple) of 48 and 54, we need to find the multiples of 48 and 54 (multiples of 48 = 48, 96, 144, 192 . . . . 432; multiples of 54 = 54, 108, 162, 216 . . . . 432) and choose the smallest multiple that is exactly divisible by 48 and 54, i.e., 432.

What are the Methods to Find LCM of 48 and 54?

The commonly used methods to find the LCM of 48 and 54 are:

  • Prime Factorization Method
  • Listing Multiples
  • Division Method

If the LCM of 54 and 48 is 432, Find its GCF.

LCM(54, 48) × GCF(54, 48) = 54 × 48
Since the LCM of 54 and 48 = 432
⇒ 432 × GCF(54, 48) = 2592
Therefore, the GCF (greatest common factor) = 2592/432 = 6.

Which of the following is the LCM of 48 and 54? 5, 432, 3, 30

The value of LCM of 48, 54 is the smallest common multiple of 48 and 54. The number satisfying the given condition is 432.

What is the Relation Between GCF and LCM of 48, 54?

The following equation can be used to express the relation between GCF and LCM of 48 and 54, i.e. GCF × LCM = 48 × 54.

 
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