LCM of 4, 6, and 12
LCM of 4, 6, and 12 is the smallest number among all common multiples of 4, 6, and 12. The first few multiples of 4, 6, and 12 are (4, 8, 12, 16, 20 . . .), (6, 12, 18, 24, 30 . . .), and (12, 24, 36, 48, 60 . . .) respectively. There are 3 commonly used methods to find LCM of 4, 6, 12  by listing multiples, by prime factorization, and by division method.
1.  LCM of 4, 6, and 12 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 4, 6, and 12?
Answer: LCM of 4, 6, and 12 is 12.
Explanation:
The LCM of three nonzero integers, a(4), b(6), and c(12), is the smallest positive integer m(12) that is divisible by a(4), b(6), and c(12) without any remainder.
Methods to Find LCM of 4, 6, and 12
Let's look at the different methods for finding the LCM of 4, 6, and 12.
 By Prime Factorization Method
 By Listing Multiples
 By Division Method
LCM of 4, 6, and 12 by Prime Factorization
Prime factorization of 4, 6, and 12 is (2 × 2) = 2^{2}, (2 × 3) = 2^{1} × 3^{1}, and (2 × 2 × 3) = 2^{2} × 3^{1} respectively. LCM of 4, 6, and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{2} × 3^{1} = 12.
Hence, the LCM of 4, 6, and 12 by prime factorization is 12.
LCM of 4, 6, and 12 by Listing Multiples
To calculate the LCM of 4, 6, 12 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 4 (4, 8, 12, 16, 20 . . .), 6 (6, 12, 18, 24, 30 . . .), and 12 (12, 24, 36, 48, 60 . . .).
 Step 2: The common multiples from the multiples of 4, 6, and 12 are 12, 24, . . .
 Step 3: The smallest common multiple of 4, 6, and 12 is 12.
∴ The least common multiple of 4, 6, and 12 = 12.
LCM of 4, 6, and 12 by Division Method
To calculate the LCM of 4, 6, and 12 by the division method, we will divide the numbers(4, 6, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 4, 6, and 12.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 4, 6, and 12. Write this prime number(2) on the left of the given numbers(4, 6, and 12), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (4, 6, 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 4, 6, and 12 is the product of all prime numbers on the left, i.e. LCM(4, 6, 12) by division method = 2 × 2 × 3 = 12.
ā Also Check:
 LCM of 18 and 54  54
 LCM of 35 and 60  420
 LCM of 2, 4, 6, 8, 10 and 12  120
 LCM of 10 and 50  50
 LCM of 25 and 36  900
 LCM of 45 and 60  180
 LCM of 56 and 70  280
LCM of 4, 6, and 12 Examples

Example 1: Calculate the LCM of 4, 6, and 12 using the GCD of the given numbers.
Solution:
Prime factorization of 4, 6, 12:
 4 = 2^{2}
 6 = 2^{1} × 3^{1}
 12 = 2^{2} × 3^{1}
Therefore, GCD(4, 6) = 2, GCD(6, 12) = 6, GCD(4, 12) = 4, GCD(4, 6, 12) = 2
We know,
LCM(4, 6, 12) = [(4 × 6 × 12) × GCD(4, 6, 12)]/[GCD(4, 6) × GCD(6, 12) × GCD(4, 12)]
LCM(4, 6, 12) = (288 × 2)/(2 × 6 × 4) = 12
⇒LCM(4, 6, 12) = 12 
Example 2: Find the smallest number that is divisible by 4, 6, 12 exactly.
Solution:
The smallest number that is divisible by 4, 6, and 12 exactly is their LCM.
⇒ Multiples of 4, 6, and 12: Multiples of 4 = 4, 8, 12, 16, 20, . . . .
 Multiples of 6 = 6, 12, 18, 24, 30, . . . .
 Multiples of 12 = 12, 24, 36, 48, 60, . . . .
Therefore, the LCM of 4, 6, and 12 is 12.

Example 3: Verify the relationship between the GCD and LCM of 4, 6, and 12.
Solution:
The relation between GCD and LCM of 4, 6, and 12 is given as,
LCM(4, 6, 12) = [(4 × 6 × 12) × GCD(4, 6, 12)]/[GCD(4, 6) × GCD(6, 12) × GCD(4, 12)]
⇒ Prime factorization of 4, 6 and 12: 4 = 2^{2}
 6 = 2^{1} × 3^{1}
 12 = 2^{2} × 3^{1}
∴ GCD of (4, 6), (6, 12), (4, 12) and (4, 6, 12) = 2, 6, 4 and 2 respectively.
Now, LHS = LCM(4, 6, 12) = 12.
And, RHS = [(4 × 6 × 12) × GCD(4, 6, 12)]/[GCD(4, 6) × GCD(6, 12) × GCD(4, 12)] = [(288) × 2]/[2 × 6 × 4] = 12
LHS = RHS = 12.
Hence verified.
FAQs on LCM of 4, 6, and 12
What is the LCM of 4, 6, and 12?
The LCM of 4, 6, and 12 is 12. To find the LCM (least common multiple) of 4, 6, and 12, we need to find the multiples of 4, 6, and 12 (multiples of 4 = 4, 8 . . . .; multiples of 6 = 6, 12 . . . .; multiples of 12 = 12, 24 . . . .) and choose the smallest multiple that is exactly divisible by 4, 6, and 12, i.e., 12.
How to Find the LCM of 4, 6, and 12 by Prime Factorization?
To find the LCM of 4, 6, and 12 using prime factorization, we will find the prime factors, (4 = 2^{2}), (6 = 2^{1} × 3^{1}), and (12 = 2^{2} × 3^{1}). LCM of 4, 6, and 12 is the product of prime factors raised to their respective highest exponent among the numbers 4, 6, and 12.
⇒ LCM of 4, 6, 12 = 2^{2} × 3^{1} = 12.
What are the Methods to Find LCM of 4, 6, 12?
The commonly used methods to find the LCM of 4, 6, 12 are:
 Division Method
 Listing Multiples
 Prime Factorization Method
Which of the following is the LCM of 4, 6, and 12? 30, 16, 12, 3
The value of LCM of 4, 6, 12 is the smallest common multiple of 4, 6, and 12. The number satisfying the given condition is 12.
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