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GCF of 6 and 21

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

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

What is GCF of 6 and 21?

Answer: GCF of 6 and 21 is 3.

Explanation:

The GCF of two non-zero integers, x(6) and y(21), is the greatest positive integer m(3) that divides both x(6) and y(21) without any remainder.

Methods to Find GCF of 6 and 21

Let's look at the different methods for finding the GCF of 6 and 21.

Using Euclid's Algorithm
Prime Factorization Method
Long Division Method

GCF of 6 and 21 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 = 21 and Y = 6

GCF(21, 6) = GCF(6, 21 mod 6) = GCF(6, 3)
GCF(6, 3) = GCF(3, 6 mod 3) = GCF(3, 0)
GCF(3, 0) = 3 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 6 and 21 is 3.

GCF of 6 and 21 by Prime Factorization

Prime factorization of 6 and 21 is (2 × 3) and (3 × 7) respectively. As visible, 6 and 21 have only one common prime factor i.e. 3. Hence, the GCF of 6 and 21 is 3.

GCF of 6 and 21 by Long Division

GCF of 6 and 21 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 21 (larger number) by 6 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (6) by the remainder (3).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (3) is the GCF of 6 and 21.

☛ Also Check:

GCF of 6 and 21 Examples

Example 1: For two numbers, GCF = 3 and LCM = 42. If one number is 6, find the other number.

Solution:

Given: GCF (x, 6) = 3 and LCM (x, 6) = 42
∵ GCF × LCM = 6 × (x)
⇒ x = (GCF × LCM)/6
⇒ x = (3 × 42)/6
⇒ x = 21
Therefore, the other number is 21.
Example 2: Find the greatest number that divides 6 and 21 exactly.

Solution:

The greatest number that divides 6 and 21 exactly is their greatest common factor, i.e. GCF of 6 and 21.
⇒ Factors of 6 and 21:
- Factors of 6 = 1, 2, 3, 6
- Factors of 21 = 1, 3, 7, 21
Therefore, the GCF of 6 and 21 is 3.
Example 3: Find the GCF of 6 and 21, if their LCM is 42.

Solution:

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

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FAQs on GCF of 6 and 21

What is the GCF of 6 and 21?

The GCF of 6 and 21 is 3. To calculate the GCF (Greatest Common Factor) of 6 and 21, we need to factor each number (factors of 6 = 1, 2, 3, 6; factors of 21 = 1, 3, 7, 21) and choose the greatest factor that exactly divides both 6 and 21, i.e., 3.

How to Find the GCF of 6 and 21 by Long Division Method?

To find the GCF of 6, 21 using long division method, 21 is divided by 6. The corresponding divisor (3) when remainder equals 0 is taken as GCF.

If the GCF of 21 and 6 is 3, Find its LCM.

GCF(21, 6) × LCM(21, 6) = 21 × 6
Since the GCF of 21 and 6 = 3
⇒ 3 × LCM(21, 6) = 126
Therefore, LCM = 42
☛ GCF Calculator

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

To find the GCF of 6 and 21, we will find the prime factorization of the given numbers, i.e. 6 = 2 × 3; 21 = 3 × 7.
⇒ Since 3 is the only common prime factor of 6 and 21. Hence, GCF (6, 21) = 3.
☛ Prime Number

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

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

What are the Methods to Find GCF of 6 and 21?

There are three commonly used methods to find the GCF of 6 and 21.

By Prime Factorization
By Euclidean Algorithm
By Long Division

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