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GCF of 54 and 60

GCF of 54 and 60 is the largest possible number that divides 54 and 60 exactly without any remainder. The factors of 54 and 60 are 1, 2, 3, 6, 9, 18, 27, 54 and 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 respectively. There are 3 commonly used methods to find the GCF of 54 and 60 - long division, prime factorization, and Euclidean algorithm.

1.	GCF of 54 and 60
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 54 and 60?

Answer: GCF of 54 and 60 is 6.

Explanation:

The GCF of two non-zero integers, x(54) and y(60), is the greatest positive integer m(6) that divides both x(54) and y(60) without any remainder.

Methods to Find GCF of 54 and 60

Let's look at the different methods for finding the GCF of 54 and 60.

Using Euclid's Algorithm
Long Division Method
Listing Common Factors

GCF of 54 and 60 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 = 60 and Y = 54

GCF(60, 54) = GCF(54, 60 mod 54) = GCF(54, 6)
GCF(54, 6) = GCF(6, 54 mod 6) = GCF(6, 0)
GCF(6, 0) = 6 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 54 and 60 is 6.

GCF of 54 and 60 by Long Division

GCF of 54 and 60 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 60 (larger number) by 54 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (54) by the remainder (6).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (6) is the GCF of 54 and 60.

GCF of 54 and 60 by Listing Common Factors

Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

There are 4 common factors of 54 and 60, that are 1, 2, 3, and 6. Therefore, the greatest common factor of 54 and 60 is 6.

☛ Also Check:

GCF of 54 and 60 Examples

Example 1: The product of two numbers is 3240. If their GCF is 6, what is their LCM?

Solution:

Given: GCF = 6 and product of numbers = 3240
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 3240/6
Therefore, the LCM is 540.
Example 2: Find the GCF of 54 and 60, if their LCM is 540.

Solution:

∵ LCM × GCF = 54 × 60
⇒ GCF(54, 60) = (54 × 60)/540 = 6
Therefore, the greatest common factor of 54 and 60 is 6.
Example 3: Find the greatest number that divides 54 and 60 exactly.

Solution:

The greatest number that divides 54 and 60 exactly is their greatest common factor, i.e. GCF of 54 and 60.
⇒ Factors of 54 and 60:
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
- Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Therefore, the GCF of 54 and 60 is 6.

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FAQs on GCF of 54 and 60

What is the GCF of 54 and 60?

The GCF of 54 and 60 is 6. To calculate the greatest common factor of 54 and 60, we need to factor each number (factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54; factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) and choose the greatest factor that exactly divides both 54 and 60, i.e., 6.

How to Find the GCF of 54 and 60 by Prime Factorization?

To find the GCF of 54 and 60, we will find the prime factorization of the given numbers, i.e. 54 = 2 × 3 × 3 × 3; 60 = 2 × 2 × 3 × 5.
⇒ Since 2, 3 are common terms in the prime factorization of 54 and 60. Hence, GCF(54, 60) = 2 × 3 = 6
☛ Prime Numbers

What is the Relation Between LCM and GCF of 54, 60?

The following equation can be used to express the relation between LCM and GCF of 54 and 60, i.e. GCF × LCM = 54 × 60.

What are the Methods to Find GCF of 54 and 60?

There are three commonly used methods to find the GCF of 54 and 60.

By Long Division
By Prime Factorization
By Euclidean Algorithm

How to Find the GCF of 54 and 60 by Long Division Method?

To find the GCF of 54, 60 using long division method, 60 is divided by 54. The corresponding divisor (6) when remainder equals 0 is taken as GCF.

If the GCF of 60 and 54 is 6, Find its LCM.

GCF(60, 54) × LCM(60, 54) = 60 × 54
Since the GCF of 60 and 54 = 6
⇒ 6 × LCM(60, 54) = 3240
Therefore, LCM = 540
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