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LCM of 3, 9, and 12

LCM of 3, 9, and 12 is the smallest number among all common multiples of 3, 9, and 12. The first few multiples of 3, 9, and 12 are (3, 6, 9, 12, 15 . . .), (9, 18, 27, 36, 45 . . .), and (12, 24, 36, 48, 60 . . .) respectively. There are 3 commonly used methods to find LCM of 3, 9, 12 - by listing multiples, by prime factorization, and by division method.

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

What is the LCM of 3, 9, and 12?

Answer: LCM of 3, 9, and 12 is 36.

Explanation:

The LCM of three non-zero integers, a(3), b(9), and c(12), is the smallest positive integer m(36) that is divisible by a(3), b(9), and c(12) without any remainder.

Methods to Find LCM of 3, 9, and 12

Let's look at the different methods for finding the LCM of 3, 9, and 12.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 3, 9, and 12 by Prime Factorization

Prime factorization of 3, 9, and 12 is (3) = 3¹, (3 × 3) = 3², and (2 × 2 × 3) = 2² × 3¹ respectively. LCM of 3, 9, and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² = 36.
Hence, the LCM of 3, 9, and 12 by prime factorization is 36.

LCM of 3, 9, and 12 by Division Method

To calculate the LCM of 3, 9, and 12 by the division method, we will divide the numbers(3, 9, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 3, 9, and 12.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 3, 9, and 12. Write this prime number(2) on the left of the given numbers(3, 9, and 12), separated as per the ladder arrangement.
Step 2: If any of the given numbers (3, 9, 12) 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 3, 9, and 12 is the product of all prime numbers on the left, i.e. LCM(3, 9, 12) by division method = 2 × 2 × 3 × 3 = 36.

LCM of 3, 9, and 12 by Listing Multiples

To calculate the LCM of 3, 9, 12 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 3 (3, 6, 9, 12, 15 . . .), 9 (9, 18, 27, 36, 45 . . .), and 12 (12, 24, 36, 48, 60 . . .).
Step 2: The common multiples from the multiples of 3, 9, and 12 are 36, 72, . . .
Step 3: The smallest common multiple of 3, 9, and 12 is 36.

∴ The least common multiple of 3, 9, and 12 = 36.

☛ Also Check:

LCM of 3, 9, and 12 Examples

Example 1: Verify the relationship between the GCD and LCM of 3, 9, and 12.

Solution:

The relation between GCD and LCM of 3, 9, and 12 is given as,
LCM(3, 9, 12) = [(3 × 9 × 12) × GCD(3, 9, 12)]/[GCD(3, 9) × GCD(9, 12) × GCD(3, 12)]
⇒ Prime factorization of 3, 9 and 12:
- 3 = 3¹
- 9 = 3²
- 12 = 2² × 3¹
∴ GCD of (3, 9), (9, 12), (3, 12) and (3, 9, 12) = 3, 3, 3 and 3 respectively.
Now, LHS = LCM(3, 9, 12) = 36.
And, RHS = [(3 × 9 × 12) × GCD(3, 9, 12)]/[GCD(3, 9) × GCD(9, 12) × GCD(3, 12)] = [(324) × 3]/[3 × 3 × 3] = 36
LHS = RHS = 36.
Hence verified.
Example 2: Calculate the LCM of 3, 9, and 12 using the GCD of the given numbers.

Solution:

Prime factorization of 3, 9, 12:
- 3 = 3¹
- 9 = 3²
- 12 = 2² × 3¹
Therefore, GCD(3, 9) = 3, GCD(9, 12) = 3, GCD(3, 12) = 3, GCD(3, 9, 12) = 3
We know,
LCM(3, 9, 12) = [(3 × 9 × 12) × GCD(3, 9, 12)]/[GCD(3, 9) × GCD(9, 12) × GCD(3, 12)]
LCM(3, 9, 12) = (324 × 3)/(3 × 3 × 3) = 36
⇒LCM(3, 9, 12) = 36
Example 3: Find the smallest number that is divisible by 3, 9, 12 exactly.

Solution:

The smallest number that is divisible by 3, 9, and 12 exactly is their LCM.
⇒ Multiples of 3, 9, and 12:
- Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, . . . .
- Multiples of 9 = 9, 18, 27, 36, 45, 54, . . . .
- Multiples of 12 = 12, 24, 36, 48, 60, 72, . . . .
Therefore, the LCM of 3, 9, and 12 is 36.

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FAQs on LCM of 3, 9, and 12

What is the LCM of 3, 9, and 12?

The LCM of 3, 9, and 12 is 36. To find the least common multiple of 3, 9, and 12, we need to find the multiples of 3, 9, and 12 (multiples of 3 = 3, 6, 9, 12 . . . . 36 . . . . ; multiples of 9 = 9, 18, 27, 36 . . . .; multiples of 12 = 12, 24, 36, 48 . . . .) and choose the smallest multiple that is exactly divisible by 3, 9, and 12, i.e., 36.

How to Find the LCM of 3, 9, and 12 by Prime Factorization?

To find the LCM of 3, 9, and 12 using prime factorization, we will find the prime factors, (3 = 3¹), (9 = 3²), and (12 = 2² × 3¹). LCM of 3, 9, and 12 is the product of prime factors raised to their respective highest exponent among the numbers 3, 9, and 12.
⇒ LCM of 3, 9, 12 = 2² × 3² = 36.

What is the Relation Between GCF and LCM of 3, 9, 12?

The following equation can be used to express the relation between GCF and LCM of 3, 9, 12, i.e. LCM(3, 9, 12) = [(3 × 9 × 12) × GCF(3, 9, 12)]/[GCF(3, 9) × GCF(9, 12) × GCF(3, 12)].

What is the Least Perfect Square Divisible by 3, 9, and 12?

The least number divisible by 3, 9, and 12 = LCM(3, 9, 12)
LCM of 3, 9, and 12 = 2 × 2 × 3 × 3 [No incomplete pair]
⇒ Least perfect square divisible by each 3, 9, and 12 = LCM(3, 9, 12) = 36 [Square root of 36 = √36 = ±6]
Therefore, 36 is the required number.

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