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GCF of 12 and 9

GCF of 12 and 9 is the largest possible number that divides 12 and 9 exactly without any remainder. The factors of 12 and 9 are 1, 2, 3, 4, 6, 12 and 1, 3, 9 respectively. There are 3 commonly used methods to find the GCF of 12 and 9 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 12 and 9
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 12 and 9?

Answer: GCF of 12 and 9 is 3.

Explanation:

The GCF of two non-zero integers, x(12) and y(9), is the greatest positive integer m(3) that divides both x(12) and y(9) without any remainder.

Methods to Find GCF of 12 and 9

Let's look at the different methods for finding the GCF of 12 and 9.

Prime Factorization Method
Long Division Method
Using Euclid's Algorithm

GCF of 12 and 9 by Prime Factorization

Prime factorization of 12 and 9 is (2 × 2 × 3) and (3 × 3) respectively. As visible, 12 and 9 have only one common prime factor i.e. 3. Hence, the GCF of 12 and 9 is 3.

GCF of 12 and 9 by Long Division

GCF of 12 and 9 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

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

The corresponding divisor (3) is the GCF of 12 and 9.

GCF of 12 and 9 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 = 12 and Y = 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 9 is 3.

☛ Also Check:

GCF of 12 and 9 Examples

Example 1: For two numbers, GCF = 3 and LCM = 36. If one number is 12, find the other number.

Solution:

Given: GCF (y, 12) = 3 and LCM (y, 12) = 36
∵ GCF × LCM = 12 × (y)
⇒ y = (GCF × LCM)/12
⇒ y = (3 × 36)/12
⇒ y = 9
Therefore, the other number is 9.
Example 2: Find the GCF of 12 and 9, if their LCM is 36.

Solution:

∵ LCM × GCF = 12 × 9
⇒ GCF(12, 9) = (12 × 9)/36 = 3
Therefore, the greatest common factor of 12 and 9 is 3.
Example 3: Find the greatest number that divides 12 and 9 exactly.

Solution:

The greatest number that divides 12 and 9 exactly is their greatest common factor, i.e. GCF of 12 and 9.
⇒ Factors of 12 and 9:
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 9 = 1, 3, 9
Therefore, the GCF of 12 and 9 is 3.

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FAQs on GCF of 12 and 9

What is the GCF of 12 and 9?

The GCF of 12 and 9 is 3. To calculate the greatest common factor of 12 and 9, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; factors of 9 = 1, 3, 9) and choose the greatest factor that exactly divides both 12 and 9, i.e., 3.

If the GCF of 9 and 12 is 3, Find its LCM.

GCF(9, 12) × LCM(9, 12) = 9 × 12
Since the GCF of 9 and 12 = 3
⇒ 3 × LCM(9, 12) = 108
Therefore, LCM = 36
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What are the Methods to Find GCF of 12 and 9?

There are three commonly used methods to find the GCF of 12 and 9.

By Prime Factorization
By Long Division
By Listing Common Factors

What is the Relation Between LCM and GCF of 12, 9?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 12 and 9, i.e. GCF × LCM = 12 × 9.

How to Find the GCF of 12 and 9 by Long Division Method?

To find the GCF of 12, 9 using long division method, 12 is divided by 9. The corresponding divisor (3) when remainder equals 0 is taken as GCF.

How to Find the GCF of 12 and 9 by Prime Factorization?

To find the GCF of 12 and 9, we will find the prime factorization of the given numbers, i.e. 12 = 2 × 2 × 3; 9 = 3 × 3.
⇒ Since 3 is the only common prime factor of 12 and 9. Hence, GCF (12, 9) = 3.
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