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GCF of 12 and 54

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

1.	GCF of 12 and 54
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 12 and 54?

Answer: GCF of 12 and 54 is 6.

Explanation:

The GCF of two non-zero integers, x(12) and y(54), is the greatest positive integer m(6) that divides both x(12) and y(54) without any remainder.

Methods to Find GCF of 12 and 54

The methods to find the GCF of 12 and 54 are explained below.

Prime Factorization Method
Long Division Method
Using Euclid's Algorithm

GCF of 12 and 54 by Prime Factorization

Prime factorization of 12 and 54 is (2 × 2 × 3) and (2 × 3 × 3 × 3) respectively. As visible, 12 and 54 have common prime factors. Hence, the GCF of 12 and 54 is 2 × 3 = 6.

GCF of 12 and 54 by Long Division

GCF of 12 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 12 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (12) by the remainder (6).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (6) is the GCF of 12 and 54.

GCF of 12 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 = 12

GCF(54, 12) = GCF(12, 54 mod 12) = GCF(12, 6)
GCF(12, 6) = GCF(6, 12 mod 6) = GCF(6, 0)
GCF(6, 0) = 6 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 12 and 54 is 6.

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GCF of 12 and 54 Examples

Example 1: Find the greatest number that divides 12 and 54 exactly.

Solution:

The greatest number that divides 12 and 54 exactly is their greatest common factor, i.e. GCF of 12 and 54.
⇒ Factors of 12 and 54:
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
Therefore, the GCF of 12 and 54 is 6.
Example 2: The product of two numbers is 648. If their GCF is 6, what is their LCM?

Solution:

Given: GCF = 6 and product of numbers = 648
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 648/6
Therefore, the LCM is 108.
Example 3: For two numbers, GCF = 6 and LCM = 108. If one number is 54, find the other number.

Solution:

Given: GCF (x, 54) = 6 and LCM (x, 54) = 108
∵ GCF × LCM = 54 × (x)
⇒ x = (GCF × LCM)/54
⇒ x = (6 × 108)/54
⇒ x = 12
Therefore, the other number is 12.

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FAQs on GCF of 12 and 54

What is the GCF of 12 and 54?

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

How to Find the GCF of 12 and 54 by Prime Factorization?

To find the GCF of 12 and 54, we will find the prime factorization of the given numbers, i.e. 12 = 2 × 2 × 3; 54 = 2 × 3 × 3 × 3.
⇒ Since 2, 3 are common terms in the prime factorization of 12 and 54. Hence, GCF(12, 54) = 2 × 3 = 6
☛ What is a Prime Number?

How to Find the GCF of 12 and 54 by Long Division Method?

To find the GCF of 12, 54 using long division method, 54 is divided by 12. The corresponding divisor (6) when remainder equals 0 is taken as GCF.

What are the Methods to Find GCF of 12 and 54?

There are three commonly used methods to find the GCF of 12 and 54.

By Euclidean Algorithm
By Prime Factorization
By Long Division

If the GCF of 54 and 12 is 6, Find its LCM.

GCF(54, 12) × LCM(54, 12) = 54 × 12
Since the GCF of 54 and 12 = 6
⇒ 6 × LCM(54, 12) = 648
Therefore, LCM = 108
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What is the Relation Between LCM and GCF of 12, 54?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 12 and 54, i.e. GCF × LCM = 12 × 54.

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