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LCM of 6, 12, and 14

LCM of 6, 12, and 14 is the smallest number among all common multiples of 6, 12, and 14. The first few multiples of 6, 12, and 14 are (6, 12, 18, 24, 30 . . .), (12, 24, 36, 48, 60 . . .), and (14, 28, 42, 56, 70 . . .) respectively. There are 3 commonly used methods to find LCM of 6, 12, 14 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 6, 12, and 14
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 6, 12, and 14?

Answer: LCM of 6, 12, and 14 is 84.

Explanation:

The LCM of three non-zero integers, a(6), b(12), and c(14), is the smallest positive integer m(84) that is divisible by a(6), b(12), and c(14) without any remainder.

Methods to Find LCM of 6, 12, and 14

The methods to find the LCM of 6, 12, and 14 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 6, 12, and 14 by Listing Multiples

To calculate the LCM of 6, 12, 14 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 . . .), 12 (12, 24, 36, 48, 60 . . .), and 14 (14, 28, 42, 56, 70 . . .).
Step 2: The common multiples from the multiples of 6, 12, and 14 are 84, 168, . . .
Step 3: The smallest common multiple of 6, 12, and 14 is 84.

∴ The least common multiple of 6, 12, and 14 = 84.

LCM of 6, 12, and 14 by Prime Factorization

Prime factorization of 6, 12, and 14 is (2 × 3) = 2¹ × 3¹, (2 × 2 × 3) = 2² × 3¹, and (2 × 7) = 2¹ × 7¹ respectively. LCM of 6, 12, and 14 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3¹ × 7¹ = 84.
Hence, the LCM of 6, 12, and 14 by prime factorization is 84.

LCM of 6, 12, and 14 by Division Method

To calculate the LCM of 6, 12, and 14 by the division method, we will divide the numbers(6, 12, 14) by their prime factors (preferably common). The product of these divisors gives the LCM of 6, 12, and 14.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6, 12, and 14. Write this prime number(2) on the left of the given numbers(6, 12, and 14), separated as per the ladder arrangement.
Step 2: If any of the given numbers (6, 12, 14) 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, 12, and 14 is the product of all prime numbers on the left, i.e. LCM(6, 12, 14) by division method = 2 × 2 × 3 × 7 = 84.

☛ Also Check:

LCM of 6, 12, and 14 Examples

Example 1: Verify the relationship between the GCD and LCM of 6, 12, and 14.

Solution:

The relation between GCD and LCM of 6, 12, and 14 is given as,
LCM(6, 12, 14) = [(6 × 12 × 14) × GCD(6, 12, 14)]/[GCD(6, 12) × GCD(12, 14) × GCD(6, 14)]
⇒ Prime factorization of 6, 12 and 14:
- 6 = 2¹ × 3¹
- 12 = 2² × 3¹
- 14 = 2¹ × 7¹
∴ GCD of (6, 12), (12, 14), (6, 14) and (6, 12, 14) = 6, 2, 2 and 2 respectively.
Now, LHS = LCM(6, 12, 14) = 84.
And, RHS = [(6 × 12 × 14) × GCD(6, 12, 14)]/[GCD(6, 12) × GCD(12, 14) × GCD(6, 14)] = [(1008) × 2]/[6 × 2 × 2] = 84
LHS = RHS = 84.
Hence verified.
Example 2: Find the smallest number that is divisible by 6, 12, 14 exactly.

Solution:

The smallest number that is divisible by 6, 12, and 14 exactly is their LCM.
⇒ Multiples of 6, 12, and 14:
- Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, . . . .
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, . . . .
- Multiples of 14 = 14, 28, 42, 56, 70, 84, 98, . . . .
Therefore, the LCM of 6, 12, and 14 is 84.
Example 3: Calculate the LCM of 6, 12, and 14 using the GCD of the given numbers.

Solution:

Prime factorization of 6, 12, 14:
- 6 = 2¹ × 3¹
- 12 = 2² × 3¹
- 14 = 2¹ × 7¹
Therefore, GCD(6, 12) = 6, GCD(12, 14) = 2, GCD(6, 14) = 2, GCD(6, 12, 14) = 2
We know,
LCM(6, 12, 14) = [(6 × 12 × 14) × GCD(6, 12, 14)]/[GCD(6, 12) × GCD(12, 14) × GCD(6, 14)]
LCM(6, 12, 14) = (1008 × 2)/(6 × 2 × 2) = 84
⇒LCM(6, 12, 14) = 84

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FAQs on LCM of 6, 12, and 14

What is the LCM of 6, 12, and 14?

The LCM of 6, 12, and 14 is 84. To find the least common multiple (LCM) of 6, 12, and 14, we need to find the multiples of 6, 12, and 14 (multiples of 6 = 6, 12, 18, 24 . . . . 84 . . . . ; multiples of 12 = 12, 24, 36, 48, 72, 84 . . . .; multiples of 14 = 14, 28, 42, 56, 84 . . . .) and choose the smallest multiple that is exactly divisible by 6, 12, and 14, i.e., 84.

What is the Least Perfect Square Divisible by 6, 12, and 14?

The least number divisible by 6, 12, and 14 = LCM(6, 12, 14)
LCM of 6, 12, and 14 = 2 × 2 × 3 × 7 [Incomplete pair(s): 3, 7]
⇒ Least perfect square divisible by each 6, 12, and 14 = LCM(6, 12, 14) × 3 × 7 = 1764 [Square root of 1764 = √1764 = ±42]
Therefore, 1764 is the required number.

Which of the following is the LCM of 6, 12, and 14? 84, 15, 32, 3

The value of LCM of 6, 12, 14 is the smallest common multiple of 6, 12, and 14. The number satisfying the given condition is 84.

What is the Relation Between GCF and LCM of 6, 12, 14?

The following equation can be used to express the relation between GCF and LCM of 6, 12, 14, i.e. LCM(6, 12, 14) = [(6 × 12 × 14) × GCF(6, 12, 14)]/[GCF(6, 12) × GCF(12, 14) × GCF(6, 14)].

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