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LCM of 3 and 12

LCM of 3 and 12 is the smallest number among all common multiples of 3 and 12. The first few multiples of 3 and 12 are (3, 6, 9, 12, 15, 18, 21, . . . ) and (12, 24, 36, 48, 60, . . . ) respectively. There are 3 commonly used methods to find LCM of 3 and 12 - by prime factorization, by listing multiples, and by division method.

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

What is the LCM of 3 and 12?

Answer: LCM of 3 and 12 is 12.

Explanation:

The LCM of two non-zero integers, x(3) and y(12), is the smallest positive integer m(12) that is divisible by both x(3) and y(12) without any remainder.

Methods to Find LCM of 3 and 12

The methods to find the LCM of 3 and 12 are explained below.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 3 and 12 by Prime Factorization

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

LCM of 3 and 12 by Listing Multiples

To calculate the LCM of 3 and 12 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 3 (3, 6, 9, 12, 15, 18, 21, . . . ) and 12 (12, 24, 36, 48, 60, . . . . )
Step 2: The common multiples from the multiples of 3 and 12 are 12, 24, . . .
Step 3: The smallest common multiple of 3 and 12 is 12.

∴ The least common multiple of 3 and 12 = 12.

LCM of 3 and 12 by Division Method

To calculate the LCM of 3 and 12 by the division method, we will divide the numbers(3, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 3 and 12.

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

☛ Also Check:

LCM of 3 and 12 Examples

Example 1: Find the smallest number that is divisible by 3 and 12 exactly.

Solution:

The smallest number that is divisible by 3 and 12 exactly is their LCM.
⇒ Multiples of 3 and 12:
- Multiples of 3 = 3, 6, 9, 12, 15, . . . .
- Multiples of 12 = 12, 24, 36, 48, 60, . . . .
Therefore, the LCM of 3 and 12 is 12.
Example 2: The product of two numbers is 36. If their GCD is 3, what is their LCM?

Solution:

Given: GCD = 3
product of numbers = 36
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 36/3
Therefore, the LCM is 12.
The probable combination for the given case is LCM(3, 12) = 12.
Example 3: Verify the relationship between GCF and LCM of 3 and 12.

Solution:

The relation between GCF and LCM of 3 and 12 is given as,
LCM(3, 12) × GCF(3, 12) = Product of 3, 12
Prime factorization of 3 and 12 is given as, 3 = (3) = 3¹ and 12 = (2 × 2 × 3) = 2² × 3¹
LCM(3, 12) = 12
GCF(3, 12) = 3
LHS = LCM(3, 12) × GCF(3, 12) = 12 × 3 = 36
RHS = Product of 3, 12 = 3 × 12 = 36
⇒ LHS = RHS = 36
Hence, verified.

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FAQs on LCM of 3 and 12

What is the LCM of 3 and 12?

The LCM of 3 and 12 is 12. To find the least common multiple of 3 and 12, we need to find the multiples of 3 and 12 (multiples of 3 = 3, 6, 9, 12; multiples of 12 = 12, 24, 36, 48) and choose the smallest multiple that is exactly divisible by 3 and 12, i.e., 12.

If the LCM of 12 and 3 is 12, Find its GCF.

LCM(12, 3) × GCF(12, 3) = 12 × 3
Since the LCM of 12 and 3 = 12
⇒ 12 × GCF(12, 3) = 36
Therefore, the greatest common factor (GCF) = 36/12 = 3.

What is the Relation Between GCF and LCM of 3, 12?

The following equation can be used to express the relation between GCF and LCM of 3 and 12, i.e. GCF × LCM = 3 × 12.

How to Find the LCM of 3 and 12 by Prime Factorization?

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

Which of the following is the LCM of 3 and 12? 15, 3, 16, 12

The value of LCM of 3, 12 is the smallest common multiple of 3 and 12. The number satisfying the given condition is 12.

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