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GCF of 21 and 84

GCF of 21 and 84 is the largest possible number that divides 21 and 84 exactly without any remainder. The factors of 21 and 84 are 1, 3, 7, 21 and 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 respectively. There are 3 commonly used methods to find the GCF of 21 and 84 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 21 and 84
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 21 and 84?

Answer: GCF of 21 and 84 is 21.

Explanation:

The GCF of two non-zero integers, x(21) and y(84), is the greatest positive integer m(21) that divides both x(21) and y(84) without any remainder.

Methods to Find GCF of 21 and 84

The methods to find the GCF of 21 and 84 are explained below.

Using Euclid's Algorithm
Prime Factorization Method
Listing Common Factors

GCF of 21 and 84 by Euclidean Algorithm

As per the Euclidean Algorithm, GCF(X, Y) = GCF(Y, X mod Y)
where X > Y and mod is the modulo operator.

Here X = 84 and Y = 21

GCF(84, 21) = GCF(21, 84 mod 21) = GCF(21, 0)
GCF(21, 0) = 21 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 21 and 84 is 21.

GCF of 21 and 84 by Prime Factorization

Prime factorization of 21 and 84 is (3 × 7) and (2 × 2 × 3 × 7) respectively. As visible, 21 and 84 have common prime factors. Hence, the GCF of 21 and 84 is 3 × 7 = 21.

GCF of 21 and 84 by Listing Common Factors

Factors of 21: 1, 3, 7, 21
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

There are 4 common factors of 21 and 84, that are 1, 3, 21, and 7. Therefore, the greatest common factor of 21 and 84 is 21.

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GCF of 21 and 84 Examples

Example 1: Find the greatest number that divides 21 and 84 exactly.

Solution:

The greatest number that divides 21 and 84 exactly is their greatest common factor, i.e. GCF of 21 and 84.
⇒ Factors of 21 and 84:
- Factors of 21 = 1, 3, 7, 21
- Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Therefore, the GCF of 21 and 84 is 21.
Example 2: For two numbers, GCF = 21 and LCM = 84. If one number is 84, find the other number.

Solution:

Given: GCF (z, 84) = 21 and LCM (z, 84) = 84
∵ GCF × LCM = 84 × (z)
⇒ z = (GCF × LCM)/84
⇒ z = (21 × 84)/84
⇒ z = 21
Therefore, the other number is 21.
Example 3: The product of two numbers is 1764. If their GCF is 21, what is their LCM?

Solution:

Given: GCF = 21 and product of numbers = 1764
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 1764/21
Therefore, the LCM is 84.

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FAQs on GCF of 21 and 84

What is the GCF of 21 and 84?

The GCF of 21 and 84 is 21. To calculate the greatest common factor of 21 and 84, we need to factor each number (factors of 21 = 1, 3, 7, 21; factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) and choose the greatest factor that exactly divides both 21 and 84, i.e., 21.

What are the Methods to Find GCF of 21 and 84?

There are three commonly used methods to find the GCF of 21 and 84.

By Prime Factorization
By Euclidean Algorithm
By Long Division

How to Find the GCF of 21 and 84 by Prime Factorization?

To find the GCF of 21 and 84, we will find the prime factorization of the given numbers, i.e. 21 = 3 × 7; 84 = 2 × 2 × 3 × 7.
⇒ Since 3, 7 are common terms in the prime factorization of 21 and 84. Hence, GCF(21, 84) = 3 × 7 = 21
☛ What is a Prime Number?

What is the Relation Between LCM and GCF of 21, 84?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 21 and 84, i.e. GCF × LCM = 21 × 84.

How to Find the GCF of 21 and 84 by Long Division Method?

To find the GCF of 21, 84 using long division method, 84 is divided by 21. The corresponding divisor (21) when remainder equals 0 is taken as GCF.

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

GCF(84, 21) × LCM(84, 21) = 84 × 21
Since the GCF of 84 and 21 = 21
⇒ 21 × LCM(84, 21) = 1764
Therefore, LCM = 84
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