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GCF of 30 and 54

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

1.	GCF of 30 and 54
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 30 and 54?

Answer: GCF of 30 and 54 is 6.

Explanation:

The GCF of two non-zero integers, x(30) and y(54), is the greatest positive integer m(6) that divides both x(30) and y(54) without any remainder.

Methods to Find GCF of 30 and 54

The methods to find the GCF of 30 and 54 are explained below.

Prime Factorization Method
Using Euclid's Algorithm
Listing Common Factors

GCF of 30 and 54 by Prime Factorization

Prime factorization of 30 and 54 is (2 × 3 × 5) and (2 × 3 × 3 × 3) respectively. As visible, 30 and 54 have common prime factors. Hence, the GCF of 30 and 54 is 2 × 3 = 6.

GCF of 30 and 54 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 = 54 and Y = 30

GCF(54, 30) = GCF(30, 54 mod 30) = GCF(30, 24)
GCF(30, 24) = GCF(24, 30 mod 24) = GCF(24, 6)
GCF(24, 6) = GCF(6, 24 mod 6) = GCF(6, 0)
GCF(6, 0) = 6 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 30 and 54 is 6.

GCF of 30 and 54 by Listing Common Factors

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

There are 4 common factors of 30 and 54, that are 1, 2, 3, and 6. Therefore, the greatest common factor of 30 and 54 is 6.

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GCF of 30 and 54 Examples

Example 1: The product of two numbers is 1620. If their GCF is 6, what is their LCM?

Solution:

Given: GCF = 6 and product of numbers = 1620
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 1620/6
Therefore, the LCM is 270.
Example 2: Find the greatest number that divides 30 and 54 exactly.

Solution:

The greatest number that divides 30 and 54 exactly is their greatest common factor, i.e. GCF of 30 and 54.
⇒ Factors of 30 and 54:
- Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
Therefore, the GCF of 30 and 54 is 6.
Example 3: For two numbers, GCF = 6 and LCM = 270. If one number is 54, find the other number.

Solution:

Given: GCF (y, 54) = 6 and LCM (y, 54) = 270
∵ GCF × LCM = 54 × (y)
⇒ y = (GCF × LCM)/54
⇒ y = (6 × 270)/54
⇒ y = 30
Therefore, the other number is 30.

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FAQs on GCF of 30 and 54

What is the GCF of 30 and 54?

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

What are the Methods to Find GCF of 30 and 54?

There are three commonly used methods to find the GCF of 30 and 54.

By Long Division
By Prime Factorization
By Listing Common Factors

If the GCF of 54 and 30 is 6, Find its LCM.

GCF(54, 30) × LCM(54, 30) = 54 × 30
Since the GCF of 54 and 30 = 6
⇒ 6 × LCM(54, 30) = 1620
Therefore, LCM = 270
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How to Find the GCF of 30 and 54 by Prime Factorization?

To find the GCF of 30 and 54, we will find the prime factorization of the given numbers, i.e. 30 = 2 × 3 × 5; 54 = 2 × 3 × 3 × 3.
⇒ Since 2, 3 are common terms in the prime factorization of 30 and 54. Hence, GCF(30, 54) = 2 × 3 = 6
☛ What is a Prime Number?

How to Find the GCF of 30 and 54 by Long Division Method?

To find the GCF of 30, 54 using long division method, 54 is divided by 30. The corresponding divisor (6) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 30, 54?

The following equation can be used to express the relation between LCM and GCF of 30 and 54, i.e. GCF × LCM = 30 × 54.

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