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GCF of 54 and 72

GCF of 54 and 72 is the largest possible number that divides 54 and 72 exactly without any remainder. The factors of 54 and 72 are 1, 2, 3, 6, 9, 18, 27, 54 and 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 respectively. There are 3 commonly used methods to find the GCF of 54 and 72 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 54 and 72
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 54 and 72?

Answer: GCF of 54 and 72 is 18.

Explanation:

The GCF of two non-zero integers, x(54) and y(72), is the greatest positive integer m(18) that divides both x(54) and y(72) without any remainder.

Methods to Find GCF of 54 and 72

The methods to find the GCF of 54 and 72 are explained below.

Prime Factorization Method
Listing Common Factors
Using Euclid's Algorithm

GCF of 54 and 72 by Prime Factorization

Prime factorization of 54 and 72 is (2 × 3 × 3 × 3) and (2 × 2 × 2 × 3 × 3) respectively. As visible, 54 and 72 have common prime factors. Hence, the GCF of 54 and 72 is 2 × 3 × 3 = 18.

GCF of 54 and 72 by Listing Common Factors

Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

There are 6 common factors of 54 and 72, that are 1, 2, 3, 6, 9, and 18. Therefore, the greatest common factor of 54 and 72 is 18.

GCF of 54 and 72 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 = 72 and Y = 54

GCF(72, 54) = GCF(54, 72 mod 54) = GCF(54, 18)
GCF(54, 18) = GCF(18, 54 mod 18) = GCF(18, 0)
GCF(18, 0) = 18 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 54 and 72 is 18.

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GCF of 54 and 72 Examples

Example 1: Find the greatest number that divides 54 and 72 exactly.

Solution:

The greatest number that divides 54 and 72 exactly is their greatest common factor, i.e. GCF of 54 and 72.
⇒ Factors of 54 and 72:
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Therefore, the GCF of 54 and 72 is 18.
Example 2: For two numbers, GCF = 18 and LCM = 216. If one number is 72, find the other number.

Solution:

Given: GCF (z, 72) = 18 and LCM (z, 72) = 216
∵ GCF × LCM = 72 × (z)
⇒ z = (GCF × LCM)/72
⇒ z = (18 × 216)/72
⇒ z = 54
Therefore, the other number is 54.
Example 3: The product of two numbers is 3888. If their GCF is 18, what is their LCM?

Solution:

Given: GCF = 18 and product of numbers = 3888
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 3888/18
Therefore, the LCM is 216.

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FAQs on GCF of 54 and 72

What is the GCF of 54 and 72?

The GCF of 54 and 72 is 18. To calculate the greatest common factor of 54 and 72, we need to factor each number (factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54; factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) and choose the greatest factor that exactly divides both 54 and 72, i.e., 18.

What are the Methods to Find GCF of 54 and 72?

There are three commonly used methods to find the GCF of 54 and 72.

By Listing Common Factors
By Long Division
By Prime Factorization

If the GCF of 72 and 54 is 18, Find its LCM.

GCF(72, 54) × LCM(72, 54) = 72 × 54
Since the GCF of 72 and 54 = 18
⇒ 18 × LCM(72, 54) = 3888
Therefore, LCM = 216
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How to Find the GCF of 54 and 72 by Long Division Method?

To find the GCF of 54, 72 using long division method, 72 is divided by 54. The corresponding divisor (18) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 54, 72?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 54 and 72, i.e. GCF × LCM = 54 × 72.

How to Find the GCF of 54 and 72 by Prime Factorization?

To find the GCF of 54 and 72, we will find the prime factorization of the given numbers, i.e. 54 = 2 × 3 × 3 × 3; 72 = 2 × 2 × 2 × 3 × 3.
⇒ Since 2, 3, 3 are common terms in the prime factorization of 54 and 72. Hence, GCF(54, 72) = 2 × 3 × 3 = 18
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