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GCF of 9 and 15

GCF of 9 and 15 is the largest possible number that divides 9 and 15 exactly without any remainder. The factors of 9 and 15 are 1, 3, 9 and 1, 3, 5, 15 respectively. There are 3 commonly used methods to find the GCF of 9 and 15 - prime factorization, Euclidean algorithm, and long division.

1.	GCF of 9 and 15
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 9 and 15?

Answer: GCF of 9 and 15 is 3.

Explanation:

The GCF of two non-zero integers, x(9) and y(15), is the greatest positive integer m(3) that divides both x(9) and y(15) without any remainder.

Methods to Find GCF of 9 and 15

The methods to find the GCF of 9 and 15 are explained below.

Long Division Method
Using Euclid's Algorithm
Prime Factorization Method

GCF of 9 and 15 by Long Division

GCF of 9 and 15 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 15 (larger number) by 9 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (9) by the remainder (6).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (3) is the GCF of 9 and 15.

GCF of 9 and 15 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 = 15 and Y = 9

GCF(15, 9) = GCF(9, 15 mod 9) = GCF(9, 6)
GCF(9, 6) = GCF(6, 9 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 9 and 15 is 3.

GCF of 9 and 15 by Prime Factorization

Prime factorization of 9 and 15 is (3 × 3) and (3 × 5) respectively. As visible, 9 and 15 have only one common prime factor i.e. 3. Hence, the GCF of 9 and 15 is 3.

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GCF of 9 and 15 Examples

Example 1: For two numbers, GCF = 3 and LCM = 45. If one number is 15, find the other number.

Solution:

Given: GCF (x, 15) = 3 and LCM (x, 15) = 45
∵ GCF × LCM = 15 × (x)
⇒ x = (GCF × LCM)/15
⇒ x = (3 × 45)/15
⇒ x = 9
Therefore, the other number is 9.
Example 2: Find the greatest number that divides 9 and 15 exactly.

Solution:

The greatest number that divides 9 and 15 exactly is their greatest common factor, i.e. GCF of 9 and 15.
⇒ Factors of 9 and 15:
- Factors of 9 = 1, 3, 9
- Factors of 15 = 1, 3, 5, 15
Therefore, the GCF of 9 and 15 is 3.
Example 3: Find the GCF of 9 and 15, if their LCM is 45.

Solution:

∵ LCM × GCF = 9 × 15
⇒ GCF(9, 15) = (9 × 15)/45 = 3
Therefore, the greatest common factor of 9 and 15 is 3.

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FAQs on GCF of 9 and 15

What is the GCF of 9 and 15?

The GCF of 9 and 15 is 3. To calculate the greatest common factor (GCF) of 9 and 15, we need to factor each number (factors of 9 = 1, 3, 9; factors of 15 = 1, 3, 5, 15) and choose the greatest factor that exactly divides both 9 and 15, i.e., 3.

What is the Relation Between LCM and GCF of 9, 15?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 9 and 15, i.e. GCF × LCM = 9 × 15.

How to Find the GCF of 9 and 15 by Prime Factorization?

To find the GCF of 9 and 15, we will find the prime factorization of the given numbers, i.e. 9 = 3 × 3; 15 = 3 × 5.
⇒ Since 3 is the only common prime factor of 9 and 15. Hence, GCF (9, 15) = 3.
☛ What are Prime Numbers?

What are the Methods to Find GCF of 9 and 15?

There are three commonly used methods to find the GCF of 9 and 15.

By Prime Factorization
By Long Division
By Euclidean Algorithm

If the GCF of 15 and 9 is 3, Find its LCM.

GCF(15, 9) × LCM(15, 9) = 15 × 9
Since the GCF of 15 and 9 = 3
⇒ 3 × LCM(15, 9) = 135
Therefore, LCM = 45
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How to Find the GCF of 9 and 15 by Long Division Method?

To find the GCF of 9, 15 using long division method, 15 is divided by 9. The corresponding divisor (3) when remainder equals 0 is taken as GCF.

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