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GCF of 30 and 75

GCF of 30 and 75 is the largest possible number that divides 30 and 75 exactly without any remainder. The factors of 30 and 75 are 1, 2, 3, 5, 6, 10, 15, 30 and 1, 3, 5, 15, 25, 75 respectively. There are 3 commonly used methods to find the GCF of 30 and 75 - long division, Euclidean algorithm, and prime factorization.

1.	GCF of 30 and 75
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 30 and 75?

Answer: GCF of 30 and 75 is 15.

Explanation:

The GCF of two non-zero integers, x(30) and y(75), is the greatest positive integer m(15) that divides both x(30) and y(75) without any remainder.

Methods to Find GCF of 30 and 75

The methods to find the GCF of 30 and 75 are explained below.

Listing Common Factors
Using Euclid's Algorithm
Prime Factorization Method

GCF of 30 and 75 by Listing Common Factors

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 75: 1, 3, 5, 15, 25, 75

There are 4 common factors of 30 and 75, that are 1, 3, 5, and 15. Therefore, the greatest common factor of 30 and 75 is 15.

GCF of 30 and 75 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 = 75 and Y = 30

GCF(75, 30) = GCF(30, 75 mod 30) = GCF(30, 15)
GCF(30, 15) = GCF(15, 30 mod 15) = GCF(15, 0)
GCF(15, 0) = 15 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 30 and 75 is 15.

GCF of 30 and 75 by Prime Factorization

Prime factorization of 30 and 75 is (2 × 3 × 5) and (3 × 5 × 5) respectively. As visible, 30 and 75 have common prime factors. Hence, the GCF of 30 and 75 is 3 × 5 = 15.

☛ Also Check:

GCF of 30 and 75 Examples

Example 1: For two numbers, GCF = 15 and LCM = 150. If one number is 30, find the other number.

Solution:

Given: GCF (x, 30) = 15 and LCM (x, 30) = 150
∵ GCF × LCM = 30 × (x)
⇒ x = (GCF × LCM)/30
⇒ x = (15 × 150)/30
⇒ x = 75
Therefore, the other number is 75.
Example 2: Find the GCF of 30 and 75, if their LCM is 150.

Solution:

∵ LCM × GCF = 30 × 75
⇒ GCF(30, 75) = (30 × 75)/150 = 15
Therefore, the greatest common factor of 30 and 75 is 15.
Example 3: Find the greatest number that divides 30 and 75 exactly.

Solution:

The greatest number that divides 30 and 75 exactly is their greatest common factor, i.e. GCF of 30 and 75.
⇒ Factors of 30 and 75:
- Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
- Factors of 75 = 1, 3, 5, 15, 25, 75
Therefore, the GCF of 30 and 75 is 15.

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FAQs on GCF of 30 and 75

What is the GCF of 30 and 75?

The GCF of 30 and 75 is 15. To calculate the GCF (Greatest Common Factor) of 30 and 75, we need to factor each number (factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30; factors of 75 = 1, 3, 5, 15, 25, 75) and choose the greatest factor that exactly divides both 30 and 75, i.e., 15.

What is the Relation Between LCM and GCF of 30, 75?

The following equation can be used to express the relation between LCM and GCF of 30 and 75, i.e. GCF × LCM = 30 × 75.

What are the Methods to Find GCF of 30 and 75?

There are three commonly used methods to find the GCF of 30 and 75.

By Listing Common Factors
By Prime Factorization
By Long Division

How to Find the GCF of 30 and 75 by Long Division Method?

To find the GCF of 30, 75 using long division method, 75 is divided by 30. The corresponding divisor (15) when remainder equals 0 is taken as GCF.

How to Find the GCF of 30 and 75 by Prime Factorization?

To find the GCF of 30 and 75, we will find the prime factorization of the given numbers, i.e. 30 = 2 × 3 × 5; 75 = 3 × 5 × 5.
⇒ Since 3, 5 are common terms in the prime factorization of 30 and 75. Hence, GCF(30, 75) = 3 × 5 = 15
☛ Prime Number

If the GCF of 75 and 30 is 15, Find its LCM.

GCF(75, 30) × LCM(75, 30) = 75 × 30
Since the GCF of 75 and 30 = 15
⇒ 15 × LCM(75, 30) = 2250
Therefore, LCM = 150
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