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GCF of 12 and 21

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

1.	GCF of 12 and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 12 and 21?

Answer: GCF of 12 and 21 is 3.

Explanation:

The GCF of two non-zero integers, x(12) and y(21), is the greatest positive integer m(3) that divides both x(12) and y(21) without any remainder.

Methods to Find GCF of 12 and 21

The methods to find the GCF of 12 and 21 are explained below.

Using Euclid's Algorithm
Long Division Method
Listing Common Factors

GCF of 12 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 = 12

GCF(21, 12) = GCF(12, 21 mod 12) = GCF(12, 9)
GCF(12, 9) = GCF(9, 12 mod 9) = GCF(9, 3)
GCF(9, 3) = GCF(3, 9 mod 3) = GCF(3, 0)
GCF(3, 0) = 3 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 12 and 21 is 3.

GCF of 12 and 21 by Long Division

GCF of 12 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 12 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (12) by the remainder (9).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (3) is the GCF of 12 and 21.

GCF of 12 and 21 by Listing Common Factors

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 21: 1, 3, 7, 21

There are 2 common factors of 12 and 21, that are 1 and 3. Therefore, the greatest common factor of 12 and 21 is 3.

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GCF of 12 and 21 Examples

Example 1: For two numbers, GCF = 3 and LCM = 84. If one number is 12, find the other number.

Solution:

Given: GCF (z, 12) = 3 and LCM (z, 12) = 84
∵ GCF × LCM = 12 × (z)
⇒ z = (GCF × LCM)/12
⇒ z = (3 × 84)/12
⇒ z = 21
Therefore, the other number is 21.
Example 2: Find the GCF of 12 and 21, if their LCM is 84.

Solution:

∵ LCM × GCF = 12 × 21
⇒ GCF(12, 21) = (12 × 21)/84 = 3
Therefore, the greatest common factor of 12 and 21 is 3.
Example 3: The product of two numbers is 252. If their GCF is 3, what is their LCM?

Solution:

Given: GCF = 3 and product of numbers = 252
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 252/3
Therefore, the LCM is 84.

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FAQs on GCF of 12 and 21

What is the GCF of 12 and 21?

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

How to Find the GCF of 12 and 21 by Long Division Method?

To find the GCF of 12, 21 using long division method, 21 is divided by 12. The corresponding divisor (3) when remainder equals 0 is taken as GCF.

If the GCF of 21 and 12 is 3, Find its LCM.

GCF(21, 12) × LCM(21, 12) = 21 × 12
Since the GCF of 21 and 12 = 3
⇒ 3 × LCM(21, 12) = 252
Therefore, LCM = 84
☛ Greatest Common Factor Calculator

How to Find the GCF of 12 and 21 by Prime Factorization?

To find the GCF of 12 and 21, we will find the prime factorization of the given numbers, i.e. 12 = 2 × 2 × 3; 21 = 3 × 7.
⇒ Since 3 is the only common prime factor of 12 and 21. Hence, GCF (12, 21) = 3.
☛ What are Prime Numbers?

What is the Relation Between LCM and GCF of 12, 21?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 12 and 21, i.e. GCF × LCM = 12 × 21.

What are the Methods to Find GCF of 12 and 21?

There are three commonly used methods to find the GCF of 12 and 21.

By Prime Factorization
By Listing Common Factors
By Long Division

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