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GCF of 54 and 81

GCF of 54 and 81 is the largest possible number that divides 54 and 81 exactly without any remainder. The factors of 54 and 81 are 1, 2, 3, 6, 9, 18, 27, 54 and 1, 3, 9, 27, 81 respectively. There are 3 commonly used methods to find the GCF of 54 and 81 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 54 and 81
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 54 and 81?

Answer: GCF of 54 and 81 is 27.

Explanation:

The GCF of two non-zero integers, x(54) and y(81), is the greatest positive integer m(27) that divides both x(54) and y(81) without any remainder.

Methods to Find GCF of 54 and 81

Let's look at the different methods for finding the GCF of 54 and 81.

Long Division Method
Prime Factorization Method
Using Euclid's Algorithm

GCF of 54 and 81 by Long Division

GCF of 54 and 81 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 81 (larger number) by 54 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (54) by the remainder (27).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (27) is the GCF of 54 and 81.

GCF of 54 and 81 by Prime Factorization

Prime factorization of 54 and 81 is (2 × 3 × 3 × 3) and (3 × 3 × 3 × 3) respectively. As visible, 54 and 81 have common prime factors. Hence, the GCF of 54 and 81 is 3 × 3 × 3 = 27.

GCF of 54 and 81 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 = 81 and Y = 54

GCF(81, 54) = GCF(54, 81 mod 54) = GCF(54, 27)
GCF(54, 27) = GCF(27, 54 mod 27) = GCF(27, 0)
GCF(27, 0) = 27 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 54 and 81 is 27.

☛ Also Check:

GCF of 54 and 81 Examples

Example 1: Find the GCF of 54 and 81, if their LCM is 162.

Solution:

∵ LCM × GCF = 54 × 81
⇒ GCF(54, 81) = (54 × 81)/162 = 27
Therefore, the greatest common factor of 54 and 81 is 27.
Example 2: For two numbers, GCF = 27 and LCM = 162. If one number is 54, find the other number.

Solution:

Given: GCF (x, 54) = 27 and LCM (x, 54) = 162
∵ GCF × LCM = 54 × (x)
⇒ x = (GCF × LCM)/54
⇒ x = (27 × 162)/54
⇒ x = 81
Therefore, the other number is 81.
Example 3: The product of two numbers is 4374. If their GCF is 27, what is their LCM?

Solution:

Given: GCF = 27 and product of numbers = 4374
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 4374/27
Therefore, the LCM is 162.

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FAQs on GCF of 54 and 81

What is the GCF of 54 and 81?

The GCF of 54 and 81 is 27. To calculate the greatest common factor of 54 and 81, we need to factor each number (factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54; factors of 81 = 1, 3, 9, 27, 81) and choose the greatest factor that exactly divides both 54 and 81, i.e., 27.

What is the Relation Between LCM and GCF of 54, 81?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 54 and 81, i.e. GCF × LCM = 54 × 81.

What are the Methods to Find GCF of 54 and 81?

There are three commonly used methods to find the GCF of 54 and 81.

By Euclidean Algorithm
By Long Division
By Prime Factorization

How to Find the GCF of 54 and 81 by Long Division Method?

To find the GCF of 54, 81 using long division method, 81 is divided by 54. The corresponding divisor (27) when remainder equals 0 is taken as GCF.

How to Find the GCF of 54 and 81 by Prime Factorization?

To find the GCF of 54 and 81, we will find the prime factorization of the given numbers, i.e. 54 = 2 × 3 × 3 × 3; 81 = 3 × 3 × 3 × 3.
⇒ Since 3, 3, 3 are common terms in the prime factorization of 54 and 81. Hence, GCF(54, 81) = 3 × 3 × 3 = 27
☛ What are Prime Numbers?

If the GCF of 81 and 54 is 27, Find its LCM.

GCF(81, 54) × LCM(81, 54) = 81 × 54
Since the GCF of 81 and 54 = 27
⇒ 27 × LCM(81, 54) = 4374
Therefore, LCM = 162
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