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LCM of 7 and 12

LCM of 7 and 12 is the smallest number among all common multiples of 7 and 12. The first few multiples of 7 and 12 are (7, 14, 21, 28, . . . ) and (12, 24, 36, 48, 60, 72, 84, . . . ) respectively. There are 3 commonly used methods to find LCM of 7 and 12 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 7 and 12
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 7 and 12?

Answer: LCM of 7 and 12 is 84.

Explanation:

The LCM of two non-zero integers, x(7) and y(12), is the smallest positive integer m(84) that is divisible by both x(7) and y(12) without any remainder.

Methods to Find LCM of 7 and 12

Let's look at the different methods for finding the LCM of 7 and 12.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 7 and 12 by Division Method

To calculate the LCM of 7 and 12 by the division method, we will divide the numbers(7, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 7 and 12.

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

LCM of 7 and 12 by Prime Factorization

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

LCM of 7 and 12 by Listing Multiples

To calculate the LCM of 7 and 12 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 7 (7, 14, 21, 28, . . . ) and 12 (12, 24, 36, 48, 60, 72, 84, . . . . )
Step 2: The common multiples from the multiples of 7 and 12 are 84, 168, . . .
Step 3: The smallest common multiple of 7 and 12 is 84.

∴ The least common multiple of 7 and 12 = 84.

☛ Also Check:

LCM of 7 and 12 Examples

Example 1: Verify the relationship between GCF and LCM of 7 and 12.

Solution:

The relation between GCF and LCM of 7 and 12 is given as,
LCM(7, 12) × GCF(7, 12) = Product of 7, 12
Prime factorization of 7 and 12 is given as, 7 = (7) = 7¹ and 12 = (2 × 2 × 3) = 2² × 3¹
LCM(7, 12) = 84
GCF(7, 12) = 1
LHS = LCM(7, 12) × GCF(7, 12) = 84 × 1 = 84
RHS = Product of 7, 12 = 7 × 12 = 84
⇒ LHS = RHS = 84
Hence, verified.
Example 2: The GCD and LCM of two numbers are 1 and 84 respectively. If one number is 12, find the other number.

Solution:

Let the other number be a.
∵ GCD × LCM = 12 × a
⇒ a = (GCD × LCM)/12
⇒ a = (1 × 84)/12
⇒ a = 7
Therefore, the other number is 7.
Example 3: The product of two numbers is 84. If their GCD is 1, what is their LCM?

Solution:

Given: GCD = 1
product of numbers = 84
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 84/1
Therefore, the LCM is 84.
The probable combination for the given case is LCM(7, 12) = 84.

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FAQs on LCM of 7 and 12

What is the LCM of 7 and 12?

The LCM of 7 and 12 is 84. To find the least common multiple (LCM) of 7 and 12, we need to find the multiples of 7 and 12 (multiples of 7 = 7, 14, 21, 28 . . . . 84; multiples of 12 = 12, 24, 36, 48 . . . . 84) and choose the smallest multiple that is exactly divisible by 7 and 12, i.e., 84.

How to Find the LCM of 7 and 12 by Prime Factorization?

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

What is the Relation Between GCF and LCM of 7, 12?

The following equation can be used to express the relation between GCF and LCM of 7 and 12, i.e. GCF × LCM = 7 × 12.

Which of the following is the LCM of 7 and 12? 21, 12, 84, 42

The value of LCM of 7, 12 is the smallest common multiple of 7 and 12. The number satisfying the given condition is 84.

If the LCM of 12 and 7 is 84, Find its GCF.

LCM(12, 7) × GCF(12, 7) = 12 × 7
Since the LCM of 12 and 7 = 84
⇒ 84 × GCF(12, 7) = 84
Therefore, the GCF = 84/84 = 1.

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