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GCF of 60 and 75

GCF of 60 and 75 is the largest possible number that divides 60 and 75 exactly without any remainder. The factors of 60 and 75 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 and 1, 3, 5, 15, 25, 75 respectively. There are 3 commonly used methods to find the GCF of 60 and 75 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 60 and 75
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 60 and 75?

Answer: GCF of 60 and 75 is 15.

Explanation:

The GCF of two non-zero integers, x(60) and y(75), is the greatest positive integer m(15) that divides both x(60) and y(75) without any remainder.

Methods to Find GCF of 60 and 75

Let's look at the different methods for finding the GCF of 60 and 75.

Using Euclid's Algorithm
Long Division Method
Listing Common Factors

GCF of 60 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 = 60

GCF(75, 60) = GCF(60, 75 mod 60) = GCF(60, 15)
GCF(60, 15) = GCF(15, 60 mod 15) = GCF(15, 0)
GCF(15, 0) = 15 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 60 and 75 is 15.

GCF of 60 and 75 by Long Division

GCF of 60 and 75 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 75 (larger number) by 60 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (60) by the remainder (15).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (15) is the GCF of 60 and 75.

GCF of 60 and 75 by Listing Common Factors

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Factors of 75: 1, 3, 5, 15, 25, 75

There are 4 common factors of 60 and 75, that are 1, 3, 5, and 15. Therefore, the greatest common factor of 60 and 75 is 15.

☛ Also Check:

GCF of 60 and 75 Examples

Example 1: Find the greatest number that divides 60 and 75 exactly.

Solution:

The greatest number that divides 60 and 75 exactly is their greatest common factor, i.e. GCF of 60 and 75.
⇒ Factors of 60 and 75:
- Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Factors of 75 = 1, 3, 5, 15, 25, 75
Therefore, the GCF of 60 and 75 is 15.
Example 2: Find the GCF of 60 and 75, if their LCM is 300.

Solution:

∵ LCM × GCF = 60 × 75
⇒ GCF(60, 75) = (60 × 75)/300 = 15
Therefore, the greatest common factor of 60 and 75 is 15.
Example 3: For two numbers, GCF = 15 and LCM = 300. If one number is 75, find the other number.

Solution:

Given: GCF (z, 75) = 15 and LCM (z, 75) = 300
∵ GCF × LCM = 75 × (z)
⇒ z = (GCF × LCM)/75
⇒ z = (15 × 300)/75
⇒ z = 60
Therefore, the other number is 60.

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FAQs on GCF of 60 and 75

What is the GCF of 60 and 75?

The GCF of 60 and 75 is 15. To calculate the GCF of 60 and 75, we need to factor each number (factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60; factors of 75 = 1, 3, 5, 15, 25, 75) and choose the greatest factor that exactly divides both 60 and 75, i.e., 15.

What are the Methods to Find GCF of 60 and 75?

There are three commonly used methods to find the GCF of 60 and 75.

By Long Division
By Euclidean Algorithm
By Prime Factorization

How to Find the GCF of 60 and 75 by Long Division Method?

To find the GCF of 60, 75 using long division method, 75 is divided by 60. The corresponding divisor (15) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 60, 75?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 60 and 75, i.e. GCF × LCM = 60 × 75.

If the GCF of 75 and 60 is 15, Find its LCM.

GCF(75, 60) × LCM(75, 60) = 75 × 60
Since the GCF of 75 and 60 = 15
⇒ 15 × LCM(75, 60) = 4500
Therefore, LCM = 300
☛ Greatest Common Factor Calculator

How to Find the GCF of 60 and 75 by Prime Factorization?

To find the GCF of 60 and 75, we will find the prime factorization of the given numbers, i.e. 60 = 2 × 2 × 3 × 5; 75 = 3 × 5 × 5.
⇒ Since 3, 5 are common terms in the prime factorization of 60 and 75. Hence, GCF(60, 75) = 3 × 5 = 15
☛ Prime Number

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