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LCM of 75 and 69

LCM of 75 and 69 is the smallest number among all common multiples of 75 and 69. The first few multiples of 75 and 69 are (75, 150, 225, 300, 375, 450, 525, . . . ) and (69, 138, 207, 276, . . . ) respectively. There are 3 commonly used methods to find LCM of 75 and 69 - by listing multiples, by prime factorization, and by division method.

1. LCM of 75 and 69
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 75 and 69?

Answer: LCM of 75 and 69 is 1725.

LCM of 75 and 69

Explanation:

The LCM of two non-zero integers, x(75) and y(69), is the smallest positive integer m(1725) that is divisible by both x(75) and y(69) without any remainder.

Methods to Find LCM of 75 and 69

The methods to find the LCM of 75 and 69 are explained below.

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

LCM of 75 and 69 by Prime Factorization

Prime factorization of 75 and 69 is (3 × 5 × 5) = 31 × 52 and (3 × 23) = 31 × 231 respectively. LCM of 75 and 69 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 31 × 52 × 231 = 1725.
Hence, the LCM of 75 and 69 by prime factorization is 1725.

LCM of 75 and 69 by Listing Multiples

To calculate the LCM of 75 and 69 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 75 (75, 150, 225, 300, 375, 450, 525, . . . ) and 69 (69, 138, 207, 276, . . . . )
  • Step 2: The common multiples from the multiples of 75 and 69 are 1725, 3450, . . .
  • Step 3: The smallest common multiple of 75 and 69 is 1725.

∴ The least common multiple of 75 and 69 = 1725.

LCM of 75 and 69 by Division Method

LCM of 75 and 69 by Division Method

To calculate the LCM of 75 and 69 by the division method, we will divide the numbers(75, 69) by their prime factors (preferably common). The product of these divisors gives the LCM of 75 and 69.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 75 and 69. Write this prime number(3) on the left of the given numbers(75 and 69), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (75, 69) is a multiple of 3, divide it by 3 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 75 and 69 is the product of all prime numbers on the left, i.e. LCM(75, 69) by division method = 3 × 5 × 5 × 23 = 1725.

☛ Also Check:

LCM of 75 and 69 Examples

  1. Example 1: The product of two numbers is 5175. If their GCD is 3, what is their LCM?

    Solution:

    Given: GCD = 3
    product of numbers = 5175
    ∵ LCM × GCD = product of numbers
    ⇒ LCM = Product/GCD = 5175/3
    Therefore, the LCM is 1725.
    The probable combination for the given case is LCM(75, 69) = 1725.

  2. Example 2: Verify the relationship between GCF and LCM of 75 and 69.

    Solution:

    The relation between GCF and LCM of 75 and 69 is given as,
    LCM(75, 69) × GCF(75, 69) = Product of 75, 69
    Prime factorization of 75 and 69 is given as, 75 = (3 × 5 × 5) = 31 × 52 and 69 = (3 × 23) = 31 × 231
    LCM(75, 69) = 1725
    GCF(75, 69) = 3
    LHS = LCM(75, 69) × GCF(75, 69) = 1725 × 3 = 5175
    RHS = Product of 75, 69 = 75 × 69 = 5175
    ⇒ LHS = RHS = 5175
    Hence, verified.

  3. Example 3: Find the smallest number that is divisible by 75 and 69 exactly.

    Solution:

    The value of LCM(75, 69) will be the smallest number that is exactly divisible by 75 and 69.
    ⇒ Multiples of 75 and 69:

    • Multiples of 75 = 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, . . . ., 1425, 1500, 1575, 1650, 1725, . . . .
    • Multiples of 69 = 69, 138, 207, 276, 345, 414, 483, 552, 621, 690, . . . ., 1449, 1518, 1587, 1656, 1725, . . . .

    Therefore, the LCM of 75 and 69 is 1725.

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FAQs on LCM of 75 and 69

What is the LCM of 75 and 69?

The LCM of 75 and 69 is 1725. To find the least common multiple of 75 and 69, we need to find the multiples of 75 and 69 (multiples of 75 = 75, 150, 225, 300 . . . . 1725; multiples of 69 = 69, 138, 207, 276 . . . . 1725) and choose the smallest multiple that is exactly divisible by 75 and 69, i.e., 1725.

What is the Least Perfect Square Divisible by 75 and 69?

The least number divisible by 75 and 69 = LCM(75, 69)
LCM of 75 and 69 = 3 × 5 × 5 × 23 [Incomplete pair(s): 3, 23]
⇒ Least perfect square divisible by each 75 and 69 = LCM(75, 69) × 3 × 23 = 119025 [Square root of 119025 = √119025 = ±345]
Therefore, 119025 is the required number.

If the LCM of 69 and 75 is 1725, Find its GCF.

LCM(69, 75) × GCF(69, 75) = 69 × 75
Since the LCM of 69 and 75 = 1725
⇒ 1725 × GCF(69, 75) = 5175
Therefore, the greatest common factor = 5175/1725 = 3.

What is the Relation Between GCF and LCM of 75, 69?

The following equation can be used to express the relation between GCF and LCM of 75 and 69, i.e. GCF × LCM = 75 × 69.

What are the Methods to Find LCM of 75 and 69?

The commonly used methods to find the LCM of 75 and 69 are:

  • Listing Multiples
  • Division Method
  • Prime Factorization Method
 
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