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LCM of 12 and 21

LCM of 12 and 21 is the smallest number among all common multiples of 12 and 21. The first few multiples of 12 and 21 are (12, 24, 36, 48, . . . ) and (21, 42, 63, 84, 105, . . . ) respectively. There are 3 commonly used methods to find LCM of 12 and 21 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 12 and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12 and 21?

Answer: LCM of 12 and 21 is 84.

Explanation:

The LCM of two non-zero integers, x(12) and y(21), is the smallest positive integer m(84) that is divisible by both x(12) and y(21) without any remainder.

Methods to Find LCM of 12 and 21

Let's look at the different methods for finding the LCM of 12 and 21.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 12 and 21 by Division Method

To calculate the LCM of 12 and 21 by the division method, we will divide the numbers(12, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 12 and 21.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12 and 21. Write this prime number(2) on the left of the given numbers(12 and 21), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 21) 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 21 is the product of all prime numbers on the left, i.e. LCM(12, 21) by division method = 2 × 2 × 3 × 7 = 84.

LCM of 12 and 21 by Prime Factorization

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

LCM of 12 and 21 by Listing Multiples

To calculate the LCM of 12 and 21 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, . . . ) and 21 (21, 42, 63, 84, 105, . . . . )
Step 2: The common multiples from the multiples of 12 and 21 are 84, 168, . . .
Step 3: The smallest common multiple of 12 and 21 is 84.

∴ The least common multiple of 12 and 21 = 84.

☛ Also Check:

LCM of 12 and 21 Examples

Example 1: Find the smallest number that is divisible by 12 and 21 exactly.

Solution:

The smallest number that is divisible by 12 and 21 exactly is their LCM.
⇒ Multiples of 12 and 21:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, . . . .
- Multiples of 21 = 21, 42, 63, 84, 105, . . . .
Therefore, the LCM of 12 and 21 is 84.
Example 2: The GCD and LCM of two numbers are 3 and 84 respectively. If one number is 12, find the other number.

Solution:

Let the other number be p.
∵ GCD × LCM = 12 × p
⇒ p = (GCD × LCM)/12
⇒ p = (3 × 84)/12
⇒ p = 21
Therefore, the other number is 21.
Example 3: Verify the relationship between GCF and LCM of 12 and 21.

Solution:

The relation between GCF and LCM of 12 and 21 is given as,
LCM(12, 21) × GCF(12, 21) = Product of 12, 21
Prime factorization of 12 and 21 is given as, 12 = (2 × 2 × 3) = 2² × 3¹ and 21 = (3 × 7) = 3¹ × 7¹
LCM(12, 21) = 84
GCF(12, 21) = 3
LHS = LCM(12, 21) × GCF(12, 21) = 84 × 3 = 252
RHS = Product of 12, 21 = 12 × 21 = 252
⇒ LHS = RHS = 252
Hence, verified.

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FAQs on LCM of 12 and 21

What is the LCM of 12 and 21?

The LCM of 12 and 21 is 84. To find the LCM (least common multiple) of 12 and 21, we need to find the multiples of 12 and 21 (multiples of 12 = 12, 24, 36, 48 . . . . 84; multiples of 21 = 21, 42, 63, 84) and choose the smallest multiple that is exactly divisible by 12 and 21, i.e., 84.

How to Find the LCM of 12 and 21 by Prime Factorization?

To find the LCM of 12 and 21 using prime factorization, we will find the prime factors, (12 = 2 × 2 × 3) and (21 = 3 × 7). LCM of 12 and 21 is the product of prime factors raised to their respective highest exponent among the numbers 12 and 21.
⇒ LCM of 12, 21 = 2² × 3¹ × 7¹ = 84.

If the LCM of 21 and 12 is 84, Find its GCF.

LCM(21, 12) × GCF(21, 12) = 21 × 12
Since the LCM of 21 and 12 = 84
⇒ 84 × GCF(21, 12) = 252
Therefore, the GCF (greatest common factor) = 252/84 = 3.

Which of the following is the LCM of 12 and 21? 21, 12, 36, 84

The value of LCM of 12, 21 is the smallest common multiple of 12 and 21. The number satisfying the given condition is 84.

What is the Relation Between GCF and LCM of 12, 21?

The following equation can be used to express the relation between GCF and LCM of 12 and 21, i.e. GCF × LCM = 12 × 21.

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