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GCF of 45 and 54

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

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

What is GCF of 45 and 54?

Answer: GCF of 45 and 54 is 9.

Explanation:

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

Methods to Find GCF of 45 and 54

Let's look at the different methods for finding the GCF of 45 and 54.

Using Euclid's Algorithm
Long Division Method
Prime Factorization Method

GCF of 45 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 = 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 45 and 54 is 9.

GCF of 45 and 54 by Long Division

GCF of 45 and 54 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 45 and 54.

GCF of 45 and 54 by Prime Factorization

Prime factorization of 45 and 54 is (3 × 3 × 5) and (2 × 3 × 3 × 3) respectively. As visible, 45 and 54 have common prime factors. Hence, the GCF of 45 and 54 is 3 × 3 = 9.

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GCF of 45 and 54 Examples

Example 1: The product of two numbers is 2430. If their GCF is 9, what is their LCM?

Solution:

Given: GCF = 9 and product of numbers = 2430
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 2430/9
Therefore, the LCM is 270.
Example 2: Find the greatest number that divides 45 and 54 exactly.

Solution:

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

Solution:

Given: GCF (z, 45) = 9 and LCM (z, 45) = 270
∵ GCF × LCM = 45 × (z)
⇒ z = (GCF × LCM)/45
⇒ z = (9 × 270)/45
⇒ z = 54
Therefore, the other number is 54.

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FAQs on GCF of 45 and 54

What is the GCF of 45 and 54?

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

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

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

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

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

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

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

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

By Prime Factorization
By Listing Common Factors
By Long Division

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

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

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