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LCM of 6, 9, and 15

LCM of 6, 9, and 15 is the smallest number among all common multiples of 6, 9, and 15. The first few multiples of 6, 9, and 15 are (6, 12, 18, 24, 30 . . .), (9, 18, 27, 36, 45 . . .), and (15, 30, 45, 60, 75 . . .) respectively. There are 3 commonly used methods to find LCM of 6, 9, 15 - by listing multiples, by prime factorization, and by division method.

1. LCM of 6, 9, and 15
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 6, 9, and 15?

Answer: LCM of 6, 9, and 15 is 90.

LCM of 6, 9, and 15

Explanation:

The LCM of three non-zero integers, a(6), b(9), and c(15), is the smallest positive integer m(90) that is divisible by a(6), b(9), and c(15) without any remainder.

Methods to Find LCM of 6, 9, and 15

Let's look at the different methods for finding the LCM of 6, 9, and 15.

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

LCM of 6, 9, and 15 by Listing Multiples

To calculate the LCM of 6, 9, 15 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 6 (6, 12, 18, 24, 30 . . .), 9 (9, 18, 27, 36, 45 . . .), and 15 (15, 30, 45, 60, 75 . . .).
  • Step 2: The common multiples from the multiples of 6, 9, and 15 are 90, 180, . . .
  • Step 3: The smallest common multiple of 6, 9, and 15 is 90.

∴ The least common multiple of 6, 9, and 15 = 90.

LCM of 6, 9, and 15 by Division Method

LCM of 6, 9, and 15 by Division Method

To calculate the LCM of 6, 9, and 15 by the division method, we will divide the numbers(6, 9, 15) by their prime factors (preferably common). The product of these divisors gives the LCM of 6, 9, and 15.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6, 9, and 15. Write this prime number(2) on the left of the given numbers(6, 9, and 15), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (6, 9, 15) 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 6, 9, and 15 is the product of all prime numbers on the left, i.e. LCM(6, 9, 15) by division method = 2 × 3 × 3 × 5 = 90.

LCM of 6, 9, and 15 by Prime Factorization

Prime factorization of 6, 9, and 15 is (2 × 3) = 21 × 31, (3 × 3) = 32, and (3 × 5) = 31 × 51 respectively. LCM of 6, 9, and 15 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 21 × 32 × 51 = 90.
Hence, the LCM of 6, 9, and 15 by prime factorization is 90.

☛ Also Check:

LCM of 6, 9, and 15 Examples

  1. Example 1: Find the smallest number that is divisible by 6, 9, 15 exactly.

    Solution:

    The smallest number that is divisible by 6, 9, and 15 exactly is their LCM.
    ⇒ Multiples of 6, 9, and 15:

    • Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, . . . .
    • Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, . . . .
    • Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, . . . .

    Therefore, the LCM of 6, 9, and 15 is 90.

  2. Example 2: Verify the relationship between the GCD and LCM of 6, 9, and 15.

    Solution:

    The relation between GCD and LCM of 6, 9, and 15 is given as,
    LCM(6, 9, 15) = [(6 × 9 × 15) × GCD(6, 9, 15)]/[GCD(6, 9) × GCD(9, 15) × GCD(6, 15)]
    ⇒ Prime factorization of 6, 9 and 15:

    • 6 = 21 × 31
    • 9 = 32
    • 15 = 31 × 51

    ∴ GCD of (6, 9), (9, 15), (6, 15) and (6, 9, 15) = 3, 3, 3 and 3 respectively.
    Now, LHS = LCM(6, 9, 15) = 90.
    And, RHS = [(6 × 9 × 15) × GCD(6, 9, 15)]/[GCD(6, 9) × GCD(9, 15) × GCD(6, 15)] = [(810) × 3]/[3 × 3 × 3] = 90
    LHS = RHS = 90.
    Hence verified.

  3. Example 3: Calculate the LCM of 6, 9, and 15 using the GCD of the given numbers.

    Solution:

    Prime factorization of 6, 9, 15:

    • 6 = 21 × 31
    • 9 = 32
    • 15 = 31 × 51

    Therefore, GCD(6, 9) = 3, GCD(9, 15) = 3, GCD(6, 15) = 3, GCD(6, 9, 15) = 3
    We know,
    LCM(6, 9, 15) = [(6 × 9 × 15) × GCD(6, 9, 15)]/[GCD(6, 9) × GCD(9, 15) × GCD(6, 15)]
    LCM(6, 9, 15) = (810 × 3)/(3 × 3 × 3) = 90
    ⇒LCM(6, 9, 15) = 90

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FAQs on LCM of 6, 9, and 15

What is the LCM of 6, 9, and 15?

The LCM of 6, 9, and 15 is 90. To find the LCM of 6, 9, and 15, we need to find the multiples of 6, 9, and 15 (multiples of 6 = 6, 12, 18, 24 . . . . 90 . . . . ; multiples of 9 = 9, 18, 27, 36 . . . . 90 . . . . ; multiples of 15 = 15, 30, 45, 60, 90 . . . .) and choose the smallest multiple that is exactly divisible by 6, 9, and 15, i.e., 90.

How to Find the LCM of 6, 9, and 15 by Prime Factorization?

To find the LCM of 6, 9, and 15 using prime factorization, we will find the prime factors, (6 = 21 × 31), (9 = 32), and (15 = 31 × 51). LCM of 6, 9, and 15 is the product of prime factors raised to their respective highest exponent among the numbers 6, 9, and 15.
⇒ LCM of 6, 9, 15 = 21 × 32 × 51 = 90.

What is the Relation Between GCF and LCM of 6, 9, 15?

The following equation can be used to express the relation between GCF and LCM of 6, 9, 15, i.e. LCM(6, 9, 15) = [(6 × 9 × 15) × GCF(6, 9, 15)]/[GCF(6, 9) × GCF(9, 15) × GCF(6, 15)].

What are the Methods to Find LCM of 6, 9, 15?

The commonly used methods to find the LCM of 6, 9, 15 are:

  • Listing Multiples
  • Prime Factorization Method
  • Division Method
 
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