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LCM of 72 and 96

LCM of 72 and 96 is the smallest number among all common multiples of 72 and 96. The first few multiples of 72 and 96 are (72, 144, 216, 288, 360, 432, 504, . . . ) and (96, 192, 288, 384, 480, 576, . . . ) respectively. There are 3 commonly used methods to find LCM of 72 and 96 - by listing multiples, by division method, and by prime factorization.

1. LCM of 72 and 96
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 72 and 96?

Answer: LCM of 72 and 96 is 288.

LCM of 72 and 96

Explanation:

The LCM of two non-zero integers, x(72) and y(96), is the smallest positive integer m(288) that is divisible by both x(72) and y(96) without any remainder.

Methods to Find LCM of 72 and 96

The methods to find the LCM of 72 and 96 are explained below.

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

LCM of 72 and 96 by Division Method

LCM of 72 and 96 by Division Method

To calculate the LCM of 72 and 96 by the division method, we will divide the numbers(72, 96) by their prime factors (preferably common). The product of these divisors gives the LCM of 72 and 96.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 72 and 96. Write this prime number(2) on the left of the given numbers(72 and 96), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (72, 96) 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 72 and 96 is the product of all prime numbers on the left, i.e. LCM(72, 96) by division method = 2 × 2 × 2 × 2 × 2 × 3 × 3 = 288.

LCM of 72 and 96 by Listing Multiples

To calculate the LCM of 72 and 96 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 72 (72, 144, 216, 288, 360, 432, 504, . . . ) and 96 (96, 192, 288, 384, 480, 576, . . . . )
  • Step 2: The common multiples from the multiples of 72 and 96 are 288, 576, . . .
  • Step 3: The smallest common multiple of 72 and 96 is 288.

∴ The least common multiple of 72 and 96 = 288.

LCM of 72 and 96 by Prime Factorization

Prime factorization of 72 and 96 is (2 × 2 × 2 × 3 × 3) = 23 × 32 and (2 × 2 × 2 × 2 × 2 × 3) = 25 × 31 respectively. LCM of 72 and 96 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 25 × 32 = 288.
Hence, the LCM of 72 and 96 by prime factorization is 288.

☛ Also Check:

LCM of 72 and 96 Examples

  1. Example 1: The GCD and LCM of two numbers are 24 and 288 respectively. If one number is 96, find the other number.

    Solution:

    Let the other number be z.
    ∵ GCD × LCM = 96 × z
    ⇒ z = (GCD × LCM)/96
    ⇒ z = (24 × 288)/96
    ⇒ z = 72
    Therefore, the other number is 72.

  2. Example 2: Verify the relationship between GCF and LCM of 72 and 96.

    Solution:

    The relation between GCF and LCM of 72 and 96 is given as,
    LCM(72, 96) × GCF(72, 96) = Product of 72, 96
    Prime factorization of 72 and 96 is given as, 72 = (2 × 2 × 2 × 3 × 3) = 23 × 32 and 96 = (2 × 2 × 2 × 2 × 2 × 3) = 25 × 31
    LCM(72, 96) = 288
    GCF(72, 96) = 24
    LHS = LCM(72, 96) × GCF(72, 96) = 288 × 24 = 6912
    RHS = Product of 72, 96 = 72 × 96 = 6912
    ⇒ LHS = RHS = 6912
    Hence, verified.

  3. Example 3: Find the smallest number that is divisible by 72 and 96 exactly.

    Solution:

    The smallest number that is divisible by 72 and 96 exactly is their LCM.
    ⇒ Multiples of 72 and 96:

    • Multiples of 72 = 72, 144, 216, 288, 360, . . . .
    • Multiples of 96 = 96, 192, 288, 384, 480, . . . .

    Therefore, the LCM of 72 and 96 is 288.

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FAQs on LCM of 72 and 96

What is the LCM of 72 and 96?

The LCM of 72 and 96 is 288. To find the LCM (least common multiple) of 72 and 96, we need to find the multiples of 72 and 96 (multiples of 72 = 72, 144, 216, 288; multiples of 96 = 96, 192, 288, 384) and choose the smallest multiple that is exactly divisible by 72 and 96, i.e., 288.

How to Find the LCM of 72 and 96 by Prime Factorization?

To find the LCM of 72 and 96 using prime factorization, we will find the prime factors, (72 = 2 × 2 × 2 × 3 × 3) and (96 = 2 × 2 × 2 × 2 × 2 × 3). LCM of 72 and 96 is the product of prime factors raised to their respective highest exponent among the numbers 72 and 96.
⇒ LCM of 72, 96 = 25 × 32 = 288.

What is the Least Perfect Square Divisible by 72 and 96?

The least number divisible by 72 and 96 = LCM(72, 96)
LCM of 72 and 96 = 2 × 2 × 2 × 2 × 2 × 3 × 3 [Incomplete pair(s): 2]
⇒ Least perfect square divisible by each 72 and 96 = LCM(72, 96) × 2 = 576 [Square root of 576 = √576 = ±24]
Therefore, 576 is the required number.

What are the Methods to Find LCM of 72 and 96?

The commonly used methods to find the LCM of 72 and 96 are:

  • Division Method
  • Prime Factorization Method
  • Listing Multiples

If the LCM of 96 and 72 is 288, Find its GCF.

LCM(96, 72) × GCF(96, 72) = 96 × 72
Since the LCM of 96 and 72 = 288
⇒ 288 × GCF(96, 72) = 6912
Therefore, the greatest common factor (GCF) = 6912/288 = 24.

 
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