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LCM of 4 and 21

LCM of 4 and 21 is the smallest number among all common multiples of 4 and 21. The first few multiples of 4 and 21 are (4, 8, 12, 16, 20, 24, . . . ) and (21, 42, 63, 84, 105, 126, . . . ) respectively. There are 3 commonly used methods to find LCM of 4 and 21 - by prime factorization, by division method, and by listing multiples.

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

What is the LCM of 4 and 21?

Answer: LCM of 4 and 21 is 84.

Explanation:

The LCM of two non-zero integers, x(4) and y(21), is the smallest positive integer m(84) that is divisible by both x(4) and y(21) without any remainder.

Methods to Find LCM of 4 and 21

Let's look at the different methods for finding the LCM of 4 and 21.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 4 and 21 by Prime Factorization

Prime factorization of 4 and 21 is (2 × 2) = 2² and (3 × 7) = 3¹ × 7¹ respectively. LCM of 4 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 4 and 21 by prime factorization is 84.

LCM of 4 and 21 by Division Method

To calculate the LCM of 4 and 21 by the division method, we will divide the numbers(4, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 4 and 21.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 4 and 21. Write this prime number(2) on the left of the given numbers(4 and 21), separated as per the ladder arrangement.
Step 2: If any of the given numbers (4, 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 4 and 21 is the product of all prime numbers on the left, i.e. LCM(4, 21) by division method = 2 × 2 × 3 × 7 = 84.

LCM of 4 and 21 by Listing Multiples

To calculate the LCM of 4 and 21 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 4 (4, 8, 12, 16, 20, 24, . . . ) and 21 (21, 42, 63, 84, 105, 126, . . . . )
Step 2: The common multiples from the multiples of 4 and 21 are 84, 168, . . .
Step 3: The smallest common multiple of 4 and 21 is 84.

∴ The least common multiple of 4 and 21 = 84.

☛ Also Check:

LCM of 4 and 21 Examples

Example 1: The GCD and LCM of two numbers are 1 and 84 respectively. If one number is 21, find the other number.

Solution:

Let the other number be p.
∵ GCD × LCM = 21 × p
⇒ p = (GCD × LCM)/21
⇒ p = (1 × 84)/21
⇒ p = 4
Therefore, the other number is 4.
Example 2: Verify the relationship between GCF and LCM of 4 and 21.

Solution:

The relation between GCF and LCM of 4 and 21 is given as,
LCM(4, 21) × GCF(4, 21) = Product of 4, 21
Prime factorization of 4 and 21 is given as, 4 = (2 × 2) = 2² and 21 = (3 × 7) = 3¹ × 7¹
LCM(4, 21) = 84
GCF(4, 21) = 1
LHS = LCM(4, 21) × GCF(4, 21) = 84 × 1 = 84
RHS = Product of 4, 21 = 4 × 21 = 84
⇒ LHS = RHS = 84
Hence, verified.
Example 3: Find the smallest number that is divisible by 4 and 21 exactly.

Solution:

The value of LCM(4, 21) will be the smallest number that is exactly divisible by 4 and 21.
⇒ Multiples of 4 and 21:
- Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, . . . ., 68, 72, 76, 80, 84, . . . .
- Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 21, 42, 63, 84, . . . .
Therefore, the LCM of 4 and 21 is 84.

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FAQs on LCM of 4 and 21

What is the LCM of 4 and 21?

The LCM of 4 and 21 is 84. To find the least common multiple (LCM) of 4 and 21, we need to find the multiples of 4 and 21 (multiples of 4 = 4, 8, 12, 16 . . . . 84; multiples of 21 = 21, 42, 63, 84) and choose the smallest multiple that is exactly divisible by 4 and 21, i.e., 84.

What is the Relation Between GCF and LCM of 4, 21?

The following equation can be used to express the relation between GCF and LCM of 4 and 21, i.e. GCF × LCM = 4 × 21.

How to Find the LCM of 4 and 21 by Prime Factorization?

To find the LCM of 4 and 21 using prime factorization, we will find the prime factors, (4 = 2 × 2) and (21 = 3 × 7). LCM of 4 and 21 is the product of prime factors raised to their respective highest exponent among the numbers 4 and 21.
⇒ LCM of 4, 21 = 2² × 3¹ × 7¹ = 84.

If the LCM of 21 and 4 is 84, Find its GCF.

LCM(21, 4) × GCF(21, 4) = 21 × 4
Since the LCM of 21 and 4 = 84
⇒ 84 × GCF(21, 4) = 84
Therefore, the GCF = 84/84 = 1.

Which of the following is the LCM of 4 and 21? 10, 32, 36, 84

The value of LCM of 4, 21 is the smallest common multiple of 4 and 21. The number satisfying the given condition is 84.

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