LCM of 9, 12, and 15 is the smallest number among all common multiples of 9, 12, and 15. The first few multiples of 9, 12, and 15 are (9, 18, 27, 36, 45 . . .), (12, 24, 36, 48, 60 . . .), and (15, 30, 45, 60, 75 . . .) respectively. There are 3 commonly used methods to find LCM of 9, 12, 15 - by division method, by listing multiples, and by prime factorization.

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1.LCM of 9, 12, and 15
2.List of Methods
3.Solved Examples
4.FAQs

Answer: LCM of 9, 12, and 15 is 180.

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Explanation:

The LCM of three non-zero integers, a(9), b(12), and c(15), is the smallest positive integer m(180) that is divisible by a(9), b(12), and c(15) without any remainder.


The methods to find the LCM of 9, 12, and 15 are explained below.

By Division MethodBy Listing MultiplesBy Prime Factorization Method

LCM of 9, 12, and 15 by Division Method

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To calculate the LCM of 9, 12, and 15 by the division method, we will divide the numbers(9, 12, 15) by their prime factors (preferably common). The product of these divisors gives the LCM of 9, 12, and 15.

Step 2: If any of the given numbers (9, 12, 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 9, 12, and 15 is the product of all prime numbers on the left, i.e. LCM(9, 12, 15) by division method = 2 × 2 × 3 × 3 × 5 = 180.

LCM of 9, 12, and 15 by Listing Multiples

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To calculate the LCM of 9, 12, 15 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 9 (9, 18, 27, 36, 45 . . .), 12 (12, 24, 36, 48, 60 . . .), and 15 (15, 30, 45, 60, 75 . . .).Step 2: The common multiples from the multiples of 9, 12, and 15 are 180, 360, . . .Step 3: The smallest common multiple of 9, 12, and 15 is 180.

∴ The least common multiple of 9, 12, and 15 = 180.

LCM of 9, 12, and 15 by Prime Factorization

Prime factorization of 9, 12, and 15 is (3 × 3) = 32, (2 × 2 × 3) = 22 × 31, and (3 × 5) = 31 × 51 respectively. LCM of 9, 12, and 15 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 32 × 51 = 180.Hence, the LCM of 9, 12, and 15 by prime factorization is 180.

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Example 1: Find the smallest number that is divisible by 9, 12, 15 exactly.

Solution:

The value of LCM(9, 12, 15) will be the smallest number that is exactly divisible by 9, 12, and 15.⇒ Multiples of 9, 12, and 15:

Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, . . . ., 153, 162, 171, 180, . . . .Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . ., 156, 168, 180, . . . .Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, . . . ., 135, 150, 165, 180, . . . .

Therefore, the LCM of 9, 12, and 15 is 180.


Example 2: Verify the relationship between the GCD and LCM of 9, 12, and 15.

Solution:

The relation between GCD and LCM of 9, 12, and 15 is given as,LCM(9, 12, 15) = <(9 × 12 × 15) × GCD(9, 12, 15)>/⇒ Prime factorization of 9, 12 and 15:

9 = 3212 = 22 × 3115 = 31 × 51

∴ GCD of (9, 12), (12, 15), (9, 15) and (9, 12, 15) = 3, 3, 3 and 3 respectively.Now, LHS = LCM(9, 12, 15) = 180.And, RHS = <(9 × 12 × 15) × GCD(9, 12, 15)>/ = <(1620) × 3>/<3 × 3 × 3> = 180LHS = RHS = 180.Hence verified.


Example 3: Calculate the LCM of 9, 12, and 15 using the GCD of the given numbers.

Solution:

Prime factorization of 9, 12, 15:

9 = 3212 = 22 × 3115 = 31 × 51

Therefore, GCD(9, 12) = 3, GCD(12, 15) = 3, GCD(9, 15) = 3, GCD(9, 12, 15) = 3We know,LCM(9, 12, 15) = <(9 × 12 × 15) × GCD(9, 12, 15)>/LCM(9, 12, 15) = (1620 × 3)/(3 × 3 × 3) = 180⇒LCM(9, 12, 15) = 180


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FAQs on LCM of 9, 12, and 15

What is the LCM of 9, 12, and 15?

The LCM of 9, 12, and 15 is 180. To find the LCM of 9, 12, and 15, we need to find the multiples of 9, 12, and 15 (multiples of 9 = 9, 18, 27, 36 . . . . 180 . . . . ; multiples of 12 = 12, 24, 36, 48 . . . . 180 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 180 . . . . ) and choose the smallest multiple that is exactly divisible by 9, 12, and 15, i.e., 180.

How to Find the LCM of 9, 12, and 15 by Prime Factorization?

To find the LCM of 9, 12, and 15 using prime factorization, we will find the prime factors, (9 = 32), (12 = 22 × 31), and (15 = 31 × 51). LCM of 9, 12, and 15 is the product of prime factors raised to their respective highest exponent among the numbers 9, 12, and 15.⇒ LCM of 9, 12, 15 = 22 × 32 × 51 = 180.

Which of the following is the LCM of 9, 12, and 15? 96, 25, 50, 180

The value of LCM of 9, 12, 15 is the smallest common multiple of 9, 12, and 15. The number satisfying the given condition is 180.

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What is the Relation Between GCF and LCM of 9, 12, 15?

The following equation can be used to express the relation between GCF and LCM of 9, 12, 15, i.e. LCM(9, 12, 15) = <(9 × 12 × 15) × GCF(9, 12, 15)>/.