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LCM of 12 and 14

LCM of 12 and 14 is the smallest number among all common multiples of 12 and 14. The first few multiples of 12 and 14 are (12, 24, 36, 48, 60, 72, 84, . . . ) and (14, 28, 42, 56, 70, 84, . . . ) respectively. There are 3 commonly used methods to find LCM of 12 and 14 - by listing multiples, by division method, and by prime factorization.

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

What is the LCM of 12 and 14?

Answer: LCM of 12 and 14 is 84.

Explanation:

The LCM of two non-zero integers, x(12) and y(14), is the smallest positive integer m(84) that is divisible by both x(12) and y(14) without any remainder.

Methods to Find LCM of 12 and 14

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

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 12 and 14 by Division Method

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

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

LCM of 12 and 14 by Prime Factorization

Prime factorization of 12 and 14 is (2 × 2 × 3) = 2² × 3¹ and (2 × 7) = 2¹ × 7¹ respectively. LCM of 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 12 and 14 by prime factorization is 84.

LCM of 12 and 14 by Listing Multiples

To calculate the LCM of 12 and 14 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 12 (12, 24, 36, 48, 60, 72, 84, . . . ) and 14 (14, 28, 42, 56, 70, 84, . . . . )
Step 2: The common multiples from the multiples of 12 and 14 are 84, 168, . . .
Step 3: The smallest common multiple of 12 and 14 is 84.

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

☛ Also Check:

LCM of 12 and 14 Examples

Example 1: Verify the relationship between GCF and LCM of 12 and 14.

Solution:

The relation between GCF and LCM of 12 and 14 is given as,
LCM(12, 14) × GCF(12, 14) = Product of 12, 14
Prime factorization of 12 and 14 is given as, 12 = (2 × 2 × 3) = 2² × 3¹ and 14 = (2 × 7) = 2¹ × 7¹
LCM(12, 14) = 84
GCF(12, 14) = 2
LHS = LCM(12, 14) × GCF(12, 14) = 84 × 2 = 168
RHS = Product of 12, 14 = 12 × 14 = 168
⇒ LHS = RHS = 168
Hence, verified.
Example 2: The product of two numbers is 168. If their GCD is 2, what is their LCM?

Solution:

Given: GCD = 2
product of numbers = 168
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 168/2
Therefore, the LCM is 84.
The probable combination for the given case is LCM(12, 14) = 84.
Example 3: The GCD and LCM of two numbers are 2 and 84 respectively. If one number is 14, find the other number.

Solution:

Let the other number be b.
∵ GCD × LCM = 14 × b
⇒ b = (GCD × LCM)/14
⇒ b = (2 × 84)/14
⇒ b = 12
Therefore, the other number is 12.

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FAQs on LCM of 12 and 14

What is the LCM of 12 and 14?

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

If the LCM of 14 and 12 is 84, Find its GCF.

LCM(14, 12) × GCF(14, 12) = 14 × 12
Since the LCM of 14 and 12 = 84
⇒ 84 × GCF(14, 12) = 168
Therefore, the greatest common factor (GCF) = 168/84 = 2.

How to Find the LCM of 12 and 14 by Prime Factorization?

To find the LCM of 12 and 14 using prime factorization, we will find the prime factors, (12 = 2 × 2 × 3) and (14 = 2 × 7). LCM of 12 and 14 is the product of prime factors raised to their respective highest exponent among the numbers 12 and 14.
⇒ LCM of 12, 14 = 2² × 3¹ × 7¹ = 84.

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

The following equation can be used to express the relation between GCF and LCM of 12 and 14, i.e. GCF × LCM = 12 × 14.

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

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

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