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GCF of 54 and 45

GCF of 54 and 45 is the largest possible number that divides 54 and 45 exactly without any remainder. The factors of 54 and 45 are 1, 2, 3, 6, 9, 18, 27, 54 and 1, 3, 5, 9, 15, 45 respectively. There are 3 commonly used methods to find the GCF of 54 and 45 - long division, prime factorization, and Euclidean algorithm.

1.	GCF of 54 and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 54 and 45?

Answer: GCF of 54 and 45 is 9.

Explanation:

The GCF of two non-zero integers, x(54) and y(45), is the greatest positive integer m(9) that divides both x(54) and y(45) without any remainder.

Methods to Find GCF of 54 and 45

The methods to find the GCF of 54 and 45 are explained below.

Listing Common Factors
Using Euclid's Algorithm
Long Division Method

GCF of 54 and 45 by Listing Common Factors

Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 45: 1, 3, 5, 9, 15, 45

There are 3 common factors of 54 and 45, that are 1, 3, and 9. Therefore, the greatest common factor of 54 and 45 is 9.

GCF of 54 and 45 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 = 45

GCF(54, 45) = GCF(45, 54 mod 45) = GCF(45, 9)
GCF(45, 9) = GCF(9, 45 mod 9) = GCF(9, 0)
GCF(9, 0) = 9 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 54 and 45 is 9.

GCF of 54 and 45 by Long Division

GCF of 54 and 45 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 54 (larger number) by 45 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (45) by the remainder (9).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (9) is the GCF of 54 and 45.

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GCF of 54 and 45 Examples

Example 1: Find the greatest number that divides 54 and 45 exactly.

Solution:

The greatest number that divides 54 and 45 exactly is their greatest common factor, i.e. GCF of 54 and 45.
⇒ Factors of 54 and 45:
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
- Factors of 45 = 1, 3, 5, 9, 15, 45
Therefore, the GCF of 54 and 45 is 9.
Example 2: For two numbers, GCF = 9 and LCM = 270. If one number is 45, find the other number.

Solution:

Given: GCF (x, 45) = 9 and LCM (x, 45) = 270
∵ GCF × LCM = 45 × (x)
⇒ x = (GCF × LCM)/45
⇒ x = (9 × 270)/45
⇒ x = 54
Therefore, the other number is 54.
Example 3: Find the GCF of 54 and 45, if their LCM is 270.

Solution:

∵ LCM × GCF = 54 × 45
⇒ GCF(54, 45) = (54 × 45)/270 = 9
Therefore, the greatest common factor of 54 and 45 is 9.

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FAQs on GCF of 54 and 45

What is the GCF of 54 and 45?

The GCF of 54 and 45 is 9. To calculate the GCF of 54 and 45, we need to factor each number (factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54; factors of 45 = 1, 3, 5, 9, 15, 45) and choose the greatest factor that exactly divides both 54 and 45, i.e., 9.

If the GCF of 45 and 54 is 9, Find its LCM.

GCF(45, 54) × LCM(45, 54) = 45 × 54
Since the GCF of 45 and 54 = 9
⇒ 9 × LCM(45, 54) = 2430
Therefore, LCM = 270
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How to Find the GCF of 54 and 45 by Prime Factorization?

To find the GCF of 54 and 45, we will find the prime factorization of the given numbers, i.e. 54 = 2 × 3 × 3 × 3; 45 = 3 × 3 × 5.
⇒ Since 3, 3 are common terms in the prime factorization of 54 and 45. Hence, GCF(54, 45) = 3 × 3 = 9
☛ What are Prime Numbers?

What are the Methods to Find GCF of 54 and 45?

There are three commonly used methods to find the GCF of 54 and 45.

By Prime Factorization
By Long Division
By Listing Common Factors

How to Find the GCF of 54 and 45 by Long Division Method?

To find the GCF of 54, 45 using long division method, 54 is divided by 45. The corresponding divisor (9) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 54, 45?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 54 and 45, i.e. GCF × LCM = 54 × 45.

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