LCM of 6, 9, and 12
LCM of 6, 9, and 12 is the smallest number among all common multiples of 6, 9, and 12. The first few multiples of 6, 9, and 12 are (6, 12, 18, 24, 30 . . .), (9, 18, 27, 36, 45 . . .), and (12, 24, 36, 48, 60 . . .) respectively. There are 3 commonly used methods to find LCM of 6, 9, 12  by listing multiples, by division method, and by prime factorization.
1.  LCM of 6, 9, and 12 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 6, 9, and 12?
Answer: LCM of 6, 9, and 12 is 36.
Explanation:
The LCM of three nonzero integers, a(6), b(9), and c(12), is the smallest positive integer m(36) that is divisible by a(6), b(9), and c(12) without any remainder.
Methods to Find LCM of 6, 9, and 12
The methods to find the LCM of 6, 9, and 12 are explained below.
 By Listing Multiples
 By Division Method
 By Prime Factorization Method
LCM of 6, 9, and 12 by Listing Multiples
To calculate the LCM of 6, 9, 12 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 12 (12, 24, 36, 48, 60 . . .).
 Step 2: The common multiples from the multiples of 6, 9, and 12 are 36, 72, . . .
 Step 3: The smallest common multiple of 6, 9, and 12 is 36.
∴ The least common multiple of 6, 9, and 12 = 36.
LCM of 6, 9, and 12 by Division Method
To calculate the LCM of 6, 9, and 12 by the division method, we will divide the numbers(6, 9, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 6, 9, and 12.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6, 9, and 12. Write this prime number(2) on the left of the given numbers(6, 9, and 12), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (6, 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 6, 9, and 12 is the product of all prime numbers on the left, i.e. LCM(6, 9, 12) by division method = 2 × 2 × 3 × 3 = 36.
LCM of 6, 9, and 12 by Prime Factorization
Prime factorization of 6, 9, and 12 is (2 × 3) = 2^{1} × 3^{1}, (3 × 3) = 3^{2}, and (2 × 2 × 3) = 2^{2} × 3^{1} respectively. LCM of 6, 9, and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{2} × 3^{2} = 36.
Hence, the LCM of 6, 9, and 12 by prime factorization is 36.
ā Also Check:
 LCM of 3 and 10  30
 LCM of 9 and 16  144
 LCM of 15 and 16  240
 LCM of 144 and 180  720
 LCM of 24 and 36  72
 LCM of 10, 12 and 15  60
 LCM of 19 and 57  57
LCM of 6, 9, and 12 Examples

Example 1: Find the smallest number that is divisible by 6, 9, 12 exactly.
Solution:
The smallest number that is divisible by 6, 9, and 12 exactly is their LCM.
⇒ Multiples of 6, 9, and 12: Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, . . . .
 Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, . . . .
 Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, . . . .
Therefore, the LCM of 6, 9, and 12 is 36.

Example 2: Verify the relationship between the GCD and LCM of 6, 9, and 12.
Solution:
The relation between GCD and LCM of 6, 9, and 12 is given as,
LCM(6, 9, 12) = [(6 × 9 × 12) × GCD(6, 9, 12)]/[GCD(6, 9) × GCD(9, 12) × GCD(6, 12)]
⇒ Prime factorization of 6, 9 and 12: 6 = 2^{1} × 3^{1}
 9 = 3^{2}
 12 = 2^{2} × 3^{1}
∴ GCD of (6, 9), (9, 12), (6, 12) and (6, 9, 12) = 3, 3, 6 and 3 respectively.
Now, LHS = LCM(6, 9, 12) = 36.
And, RHS = [(6 × 9 × 12) × GCD(6, 9, 12)]/[GCD(6, 9) × GCD(9, 12) × GCD(6, 12)] = [(648) × 3]/[3 × 3 × 6] = 36
LHS = RHS = 36.
Hence verified. 
Example 3: Calculate the LCM of 6, 9, and 12 using the GCD of the given numbers.
Solution:
Prime factorization of 6, 9, 12:
 6 = 2^{1} × 3^{1}
 9 = 3^{2}
 12 = 2^{2} × 3^{1}
Therefore, GCD(6, 9) = 3, GCD(9, 12) = 3, GCD(6, 12) = 6, GCD(6, 9, 12) = 3
We know,
LCM(6, 9, 12) = [(6 × 9 × 12) × GCD(6, 9, 12)]/[GCD(6, 9) × GCD(9, 12) × GCD(6, 12)]
LCM(6, 9, 12) = (648 × 3)/(3 × 3 × 6) = 36
⇒LCM(6, 9, 12) = 36
FAQs on LCM of 6, 9, and 12
What is the LCM of 6, 9, and 12?
The LCM of 6, 9, and 12 is 36. To find the least common multiple of 6, 9, and 12, we need to find the multiples of 6, 9, and 12 (multiples of 6 = 6, 12, 18, 24, 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 6, 9, and 12, i.e., 36.
What is the Relation Between GCF and LCM of 6, 9, 12?
The following equation can be used to express the relation between GCF and LCM of 6, 9, 12, i.e. LCM(6, 9, 12) = [(6 × 9 × 12) × GCF(6, 9, 12)]/[GCF(6, 9) × GCF(9, 12) × GCF(6, 12)].
What are the Methods to Find LCM of 6, 9, 12?
The commonly used methods to find the LCM of 6, 9, 12 are:
 Listing Multiples
 Division Method
 Prime Factorization Method
What is the Least Perfect Square Divisible by 6, 9, and 12?
The least number divisible by 6, 9, and 12 = LCM(6, 9, 12)
LCM of 6, 9, and 12 = 2 × 2 × 3 × 3 [No incomplete pair]
⇒ Least perfect square divisible by each 6, 9, and 12 = LCM(6, 9, 12) = 36 [Square root of 36 = √36 = ±6]
Therefore, 36 is the required number.
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