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LCM of 6 and 21

LCM of 6 and 21 is the smallest number among all common multiples of 6 and 21. The first few multiples of 6 and 21 are (6, 12, 18, 24, 30, 36, . . . ) and (21, 42, 63, 84, 105, . . . ) respectively. There are 3 commonly used methods to find LCM of 6 and 21 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 6 and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 6 and 21?

Answer: LCM of 6 and 21 is 42.

Explanation:

The LCM of two non-zero integers, x(6) and y(21), is the smallest positive integer m(42) that is divisible by both x(6) and y(21) without any remainder.

Methods to Find LCM of 6 and 21

The methods to find the LCM of 6 and 21 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 6 and 21 by Listing Multiples

To calculate the LCM of 6 and 21 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 6 (6, 12, 18, 24, 30, 36, . . . ) and 21 (21, 42, 63, 84, 105, . . . . )
Step 2: The common multiples from the multiples of 6 and 21 are 42, 84, . . .
Step 3: The smallest common multiple of 6 and 21 is 42.

∴ The least common multiple of 6 and 21 = 42.

LCM of 6 and 21 by Prime Factorization

Prime factorization of 6 and 21 is (2 × 3) = 2¹ × 3¹ and (3 × 7) = 3¹ × 7¹ respectively. LCM of 6 and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3¹ × 7¹ = 42.
Hence, the LCM of 6 and 21 by prime factorization is 42.

LCM of 6 and 21 by Division Method

To calculate the LCM of 6 and 21 by the division method, we will divide the numbers(6, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 6 and 21.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6 and 21. Write this prime number(2) on the left of the given numbers(6 and 21), separated as per the ladder arrangement.
Step 2: If any of the given numbers (6, 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 6 and 21 is the product of all prime numbers on the left, i.e. LCM(6, 21) by division method = 2 × 3 × 7 = 42.

☛ Also Check:

LCM of 6 and 21 Examples

Example 1: Verify the relationship between GCF and LCM of 6 and 21.

Solution:

The relation between GCF and LCM of 6 and 21 is given as,
LCM(6, 21) × GCF(6, 21) = Product of 6, 21
Prime factorization of 6 and 21 is given as, 6 = (2 × 3) = 2¹ × 3¹ and 21 = (3 × 7) = 3¹ × 7¹
LCM(6, 21) = 42
GCF(6, 21) = 3
LHS = LCM(6, 21) × GCF(6, 21) = 42 × 3 = 126
RHS = Product of 6, 21 = 6 × 21 = 126
⇒ LHS = RHS = 126
Hence, verified.
Example 2: The product of two numbers is 126. If their GCD is 3, what is their LCM?

Solution:

Given: GCD = 3
product of numbers = 126
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 126/3
Therefore, the LCM is 42.
The probable combination for the given case is LCM(6, 21) = 42.
Example 3: The GCD and LCM of two numbers are 3 and 42 respectively. If one number is 21, find the other number.

Solution:

Let the other number be y.
∵ GCD × LCM = 21 × y
⇒ y = (GCD × LCM)/21
⇒ y = (3 × 42)/21
⇒ y = 6
Therefore, the other number is 6.

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FAQs on LCM of 6 and 21

What is the LCM of 6 and 21?

The LCM of 6 and 21 is 42. To find the least common multiple of 6 and 21, we need to find the multiples of 6 and 21 (multiples of 6 = 6, 12, 18, 24 . . . . 42; multiples of 21 = 21, 42, 63, 84) and choose the smallest multiple that is exactly divisible by 6 and 21, i.e., 42.

What is the Relation Between GCF and LCM of 6, 21?

The following equation can be used to express the relation between GCF and LCM of 6 and 21, i.e. GCF × LCM = 6 × 21.

How to Find the LCM of 6 and 21 by Prime Factorization?

To find the LCM of 6 and 21 using prime factorization, we will find the prime factors, (6 = 2 × 3) and (21 = 3 × 7). LCM of 6 and 21 is the product of prime factors raised to their respective highest exponent among the numbers 6 and 21.
⇒ LCM of 6, 21 = 2¹ × 3¹ × 7¹ = 42.

Which of the following is the LCM of 6 and 21? 20, 42, 30, 12

The value of LCM of 6, 21 is the smallest common multiple of 6 and 21. The number satisfying the given condition is 42.

If the LCM of 21 and 6 is 42, Find its GCF.

LCM(21, 6) × GCF(21, 6) = 21 × 6
Since the LCM of 21 and 6 = 42
⇒ 42 × GCF(21, 6) = 126
Therefore, the GCF (greatest common factor) = 126/42 = 3.

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