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LCM of 21 and 42

LCM of 21 and 42 is the smallest number among all common multiples of 21 and 42. The first few multiples of 21 and 42 are (21, 42, 63, 84, 105, 126, . . . ) and (42, 84, 126, 168, . . . ) respectively. There are 3 commonly used methods to find LCM of 21 and 42 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 21 and 42
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 21 and 42?

Answer: LCM of 21 and 42 is 42.

Explanation:

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

Methods to Find LCM of 21 and 42

The methods to find the LCM of 21 and 42 are explained below.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 21 and 42 by Prime Factorization

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

LCM of 21 and 42 by Listing Multiples

To calculate the LCM of 21 and 42 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 21 (21, 42, 63, 84, 105, 126, . . . ) and 42 (42, 84, 126, 168, . . . . )
Step 2: The common multiples from the multiples of 21 and 42 are 42, 84, . . .
Step 3: The smallest common multiple of 21 and 42 is 42.

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

LCM of 21 and 42 by Division Method

To calculate the LCM of 21 and 42 by the division method, we will divide the numbers(21, 42) by their prime factors (preferably common). The product of these divisors gives the LCM of 21 and 42.

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

☛ Also Check:

LCM of 21 and 42 Examples

Example 1: The GCD and LCM of two numbers are 21 and 42 respectively. If one number is 21, find the other number.

Solution:

Let the other number be z.
∵ GCD × LCM = 21 × z
⇒ z = (GCD × LCM)/21
⇒ z = (21 × 42)/21
⇒ z = 42
Therefore, the other number is 42.
Example 2: The product of two numbers is 882. If their GCD is 21, what is their LCM?

Solution:

Given: GCD = 21
product of numbers = 882
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 882/21
Therefore, the LCM is 42.
The probable combination for the given case is LCM(21, 42) = 42.
Example 3: Verify the relationship between GCF and LCM of 21 and 42.

Solution:

The relation between GCF and LCM of 21 and 42 is given as,
LCM(21, 42) × GCF(21, 42) = Product of 21, 42
Prime factorization of 21 and 42 is given as, 21 = (3 × 7) = 3¹ × 7¹ and 42 = (2 × 3 × 7) = 2¹ × 3¹ × 7¹
LCM(21, 42) = 42
GCF(21, 42) = 21
LHS = LCM(21, 42) × GCF(21, 42) = 42 × 21 = 882
RHS = Product of 21, 42 = 21 × 42 = 882
⇒ LHS = RHS = 882
Hence, verified.

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FAQs on LCM of 21 and 42

What is the LCM of 21 and 42?

The LCM of 21 and 42 is 42. To find the least common multiple of 21 and 42, we need to find the multiples of 21 and 42 (multiples of 21 = 21, 42, 63, 84; multiples of 42 = 42, 84, 126, 168) and choose the smallest multiple that is exactly divisible by 21 and 42, i.e., 42.

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

LCM(42, 21) × GCF(42, 21) = 42 × 21
Since the LCM of 42 and 21 = 42
⇒ 42 × GCF(42, 21) = 882
Therefore, the greatest common factor = 882/42 = 21.

What are the Methods to Find LCM of 21 and 42?

The commonly used methods to find the LCM of 21 and 42 are:

Division Method
Listing Multiples
Prime Factorization Method

What is the Relation Between GCF and LCM of 21, 42?

The following equation can be used to express the relation between GCF and LCM of 21 and 42, i.e. GCF × LCM = 21 × 42.

What is the Least Perfect Square Divisible by 21 and 42?

The least number divisible by 21 and 42 = LCM(21, 42)
LCM of 21 and 42 = 2 × 3 × 7 [Incomplete pair(s): 2, 3, 7]
⇒ Least perfect square divisible by each 21 and 42 = LCM(21, 42) × 2 × 3 × 7 = 1764 [Square root of 1764 = √1764 = ±42]
Therefore, 1764 is the required number.

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