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LCM of 24, 36, and 48

LCM of 24, 36, and 48 is the smallest number among all common multiples of 24, 36, and 48. The first few multiples of 24, 36, and 48 are (24, 48, 72, 96, 120 . . .), (36, 72, 108, 144, 180 . . .), and (48, 96, 144, 192, 240 . . .) respectively. There are 3 commonly used methods to find LCM of 24, 36, 48 - by prime factorization, by division method, and by listing multiples.

1. LCM of 24, 36, and 48
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 24, 36, and 48?

Answer: LCM of 24, 36, and 48 is 144.

LCM of 24, 36, and 48

Explanation:

The LCM of three non-zero integers, a(24), b(36), and c(48), is the smallest positive integer m(144) that is divisible by a(24), b(36), and c(48) without any remainder.

Methods to Find LCM of 24, 36, and 48

Let's look at the different methods for finding the LCM of 24, 36, and 48.

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

LCM of 24, 36, and 48 by Prime Factorization

Prime factorization of 24, 36, and 48 is (2 × 2 × 2 × 3) = 23 × 31, (2 × 2 × 3 × 3) = 22 × 32, and (2 × 2 × 2 × 2 × 3) = 24 × 31 respectively. LCM of 24, 36, and 48 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 24 × 32 = 144.
Hence, the LCM of 24, 36, and 48 by prime factorization is 144.

LCM of 24, 36, and 48 by Listing Multiples

To calculate the LCM of 24, 36, 48 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 24 (24, 48, 72, 96, 120 . . .), 36 (36, 72, 108, 144, 180 . . .), and 48 (48, 96, 144, 192, 240 . . .).
  • Step 2: The common multiples from the multiples of 24, 36, and 48 are 144, 288, . . .
  • Step 3: The smallest common multiple of 24, 36, and 48 is 144.

∴ The least common multiple of 24, 36, and 48 = 144.

LCM of 24, 36, and 48 by Division Method

LCM of 24, 36, and 48 by Division Method

To calculate the LCM of 24, 36, and 48 by the division method, we will divide the numbers(24, 36, 48) by their prime factors (preferably common). The product of these divisors gives the LCM of 24, 36, and 48.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 24, 36, and 48. Write this prime number(2) on the left of the given numbers(24, 36, and 48), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (24, 36, 48) 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 24, 36, and 48 is the product of all prime numbers on the left, i.e. LCM(24, 36, 48) by division method = 2 × 2 × 2 × 2 × 3 × 3 = 144.

☛ Also Check:

LCM of 24, 36, and 48 Examples

  1. Example 1: Verify the relationship between the GCD and LCM of 24, 36, and 48.

    Solution:

    The relation between GCD and LCM of 24, 36, and 48 is given as,
    LCM(24, 36, 48) = [(24 × 36 × 48) × GCD(24, 36, 48)]/[GCD(24, 36) × GCD(36, 48) × GCD(24, 48)]
    ⇒ Prime factorization of 24, 36 and 48:

    • 24 = 23 × 31
    • 36 = 22 × 32
    • 48 = 24 × 31

    ∴ GCD of (24, 36), (36, 48), (24, 48) and (24, 36, 48) = 12, 12, 24 and 12 respectively.
    Now, LHS = LCM(24, 36, 48) = 144.
    And, RHS = [(24 × 36 × 48) × GCD(24, 36, 48)]/[GCD(24, 36) × GCD(36, 48) × GCD(24, 48)] = [(41472) × 12]/[12 × 12 × 24] = 144
    LHS = RHS = 144.
    Hence verified.

  2. Example 2: Find the smallest number that is divisible by 24, 36, 48 exactly.

    Solution:

    The smallest number that is divisible by 24, 36, and 48 exactly is their LCM.
    ⇒ Multiples of 24, 36, and 48:

    • Multiples of 24 = 24, 48, 72, 96, 120, 144, . . . .
    • Multiples of 36 = 36, 72, 108, 144, 180, . . . .
    • Multiples of 48 = 48, 96, 144, 192, 240, . . . .

    Therefore, the LCM of 24, 36, and 48 is 144.

  3. Example 3: Calculate the LCM of 24, 36, and 48 using the GCD of the given numbers.

    Solution:

    Prime factorization of 24, 36, 48:

    • 24 = 23 × 31
    • 36 = 22 × 32
    • 48 = 24 × 31

    Therefore, GCD(24, 36) = 12, GCD(36, 48) = 12, GCD(24, 48) = 24, GCD(24, 36, 48) = 12
    We know,
    LCM(24, 36, 48) = [(24 × 36 × 48) × GCD(24, 36, 48)]/[GCD(24, 36) × GCD(36, 48) × GCD(24, 48)]
    LCM(24, 36, 48) = (41472 × 12)/(12 × 12 × 24) = 144
    ⇒LCM(24, 36, 48) = 144

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FAQs on LCM of 24, 36, and 48

What is the LCM of 24, 36, and 48?

The LCM of 24, 36, and 48 is 144. To find the least common multiple (LCM) of 24, 36, and 48, we need to find the multiples of 24, 36, and 48 (multiples of 24 = 24, 48, 72, 96, 144 . . . .; multiples of 36 = 36, 72, 108, 144 . . . .; multiples of 48 = 48, 96, 144, 192 . . . .) and choose the smallest multiple that is exactly divisible by 24, 36, and 48, i.e., 144.

What are the Methods to Find LCM of 24, 36, 48?

The commonly used methods to find the LCM of 24, 36, 48 are:

  • Division Method
  • Listing Multiples
  • Prime Factorization Method

Which of the following is the LCM of 24, 36, and 48? 120, 2, 30, 144

The value of LCM of 24, 36, 48 is the smallest common multiple of 24, 36, and 48. The number satisfying the given condition is 144.

What is the Relation Between GCF and LCM of 24, 36, 48?

The following equation can be used to express the relation between GCF and LCM of 24, 36, 48, i.e. LCM(24, 36, 48) = [(24 × 36 × 48) × GCF(24, 36, 48)]/[GCF(24, 36) × GCF(36, 48) × GCF(24, 48)].

 
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