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LCM of 3 and 21

LCM of 3 and 21 is the smallest number among all common multiples of 3 and 21. The first few multiples of 3 and 21 are (3, 6, 9, 12, 15, 18, 21, . . . ) and (21, 42, 63, 84, 105, . . . ) respectively. There are 3 commonly used methods to find LCM of 3 and 21 - by prime factorization, by division method, and by listing multiples.

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

What is the LCM of 3 and 21?

Answer: LCM of 3 and 21 is 21.

Explanation:

The LCM of two non-zero integers, x(3) and y(21), is the smallest positive integer m(21) that is divisible by both x(3) and y(21) without any remainder.

Methods to Find LCM of 3 and 21

The methods to find the LCM of 3 and 21 are explained below.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 3 and 21 by Prime Factorization

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

LCM of 3 and 21 by Listing Multiples

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

∴ The least common multiple of 3 and 21 = 21.

LCM of 3 and 21 by Division Method

To calculate the LCM of 3 and 21 by the division method, we will divide the numbers(3, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 3 and 21.

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

☛ Also Check:

LCM of 3 and 21 Examples

Example 1: The product of two numbers is 63. If their GCD is 3, what is their LCM?

Solution:

Given: GCD = 3
product of numbers = 63
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 63/3
Therefore, the LCM is 21.
The probable combination for the given case is LCM(3, 21) = 21.
Example 2: Find the smallest number that is divisible by 3 and 21 exactly.

Solution:

The smallest number that is divisible by 3 and 21 exactly is their LCM.
⇒ Multiples of 3 and 21:
- Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, . . . .
- Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, . . . .
Therefore, the LCM of 3 and 21 is 21.
Example 3: Verify the relationship between GCF and LCM of 3 and 21.

Solution:

The relation between GCF and LCM of 3 and 21 is given as,
LCM(3, 21) × GCF(3, 21) = Product of 3, 21
Prime factorization of 3 and 21 is given as, 3 = (3) = 3¹ and 21 = (3 × 7) = 3¹ × 7¹
LCM(3, 21) = 21
GCF(3, 21) = 3
LHS = LCM(3, 21) × GCF(3, 21) = 21 × 3 = 63
RHS = Product of 3, 21 = 3 × 21 = 63
⇒ LHS = RHS = 63
Hence, verified.

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FAQs on LCM of 3 and 21

What is the LCM of 3 and 21?

The LCM of 3 and 21 is 21. To find the least common multiple (LCM) of 3 and 21, we need to find the multiples of 3 and 21 (multiples of 3 = 3, 6, 9, 12 . . . . 21; multiples of 21 = 21, 42, 63, 84) and choose the smallest multiple that is exactly divisible by 3 and 21, i.e., 21.

What is the Relation Between GCF and LCM of 3, 21?

The following equation can be used to express the relation between GCF and LCM of 3 and 21, i.e. GCF × LCM = 3 × 21.

If the LCM of 21 and 3 is 21, Find its GCF.

LCM(21, 3) × GCF(21, 3) = 21 × 3
Since the LCM of 21 and 3 = 21
⇒ 21 × GCF(21, 3) = 63
Therefore, the greatest common factor = 63/21 = 3.

How to Find the LCM of 3 and 21 by Prime Factorization?

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

Which of the following is the LCM of 3 and 21? 12, 24, 21, 42

The value of LCM of 3, 21 is the smallest common multiple of 3 and 21. The number satisfying the given condition is 21.

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