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GCF of 84 and 108

GCF of 84 and 108 is the largest possible number that divides 84 and 108 exactly without any remainder. The factors of 84 and 108 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 respectively. There are 3 commonly used methods to find the GCF of 84 and 108 - Euclidean algorithm, long division, and prime factorization.

1.	GCF of 84 and 108
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 84 and 108?

Answer: GCF of 84 and 108 is 12.

Explanation:

The GCF of two non-zero integers, x(84) and y(108), is the greatest positive integer m(12) that divides both x(84) and y(108) without any remainder.

Methods to Find GCF of 84 and 108

Let's look at the different methods for finding the GCF of 84 and 108.

Long Division Method
Listing Common Factors
Prime Factorization Method

GCF of 84 and 108 by Long Division

GCF of 84 and 108 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 108 (larger number) by 84 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (24).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the GCF of 84 and 108.

GCF of 84 and 108 by Listing Common Factors

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

There are 6 common factors of 84 and 108, that are 1, 2, 3, 4, 6, and 12. Therefore, the greatest common factor of 84 and 108 is 12.

GCF of 84 and 108 by Prime Factorization

Prime factorization of 84 and 108 is (2 × 2 × 3 × 7) and (2 × 2 × 3 × 3 × 3) respectively. As visible, 84 and 108 have common prime factors. Hence, the GCF of 84 and 108 is 2 × 2 × 3 = 12.

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GCF of 84 and 108 Examples

Example 1: Find the greatest number that divides 84 and 108 exactly.

Solution:

The greatest number that divides 84 and 108 exactly is their greatest common factor, i.e. GCF of 84 and 108.
⇒ Factors of 84 and 108:
- Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Factors of 108 = 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Therefore, the GCF of 84 and 108 is 12.
Example 2: Find the GCF of 84 and 108, if their LCM is 756.

Solution:

∵ LCM × GCF = 84 × 108
⇒ GCF(84, 108) = (84 × 108)/756 = 12
Therefore, the greatest common factor of 84 and 108 is 12.
Example 3: The product of two numbers is 9072. If their GCF is 12, what is their LCM?

Solution:

Given: GCF = 12 and product of numbers = 9072
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 9072/12
Therefore, the LCM is 756.

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FAQs on GCF of 84 and 108

What is the GCF of 84 and 108?

The GCF of 84 and 108 is 12. To calculate the greatest common factor of 84 and 108, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 108 = 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108) and choose the greatest factor that exactly divides both 84 and 108, i.e., 12.

If the GCF of 108 and 84 is 12, Find its LCM.

GCF(108, 84) × LCM(108, 84) = 108 × 84
Since the GCF of 108 and 84 = 12
⇒ 12 × LCM(108, 84) = 9072
Therefore, LCM = 756
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What are the Methods to Find GCF of 84 and 108?

There are three commonly used methods to find the GCF of 84 and 108.

By Prime Factorization
By Listing Common Factors
By Long Division

What is the Relation Between LCM and GCF of 84, 108?

The following equation can be used to express the relation between LCM and GCF of 84 and 108, i.e. GCF × LCM = 84 × 108.

How to Find the GCF of 84 and 108 by Long Division Method?

To find the GCF of 84, 108 using long division method, 108 is divided by 84. The corresponding divisor (12) when remainder equals 0 is taken as GCF.

How to Find the GCF of 84 and 108 by Prime Factorization?

To find the GCF of 84 and 108, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 108 = 2 × 2 × 3 × 3 × 3.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 84 and 108. Hence, GCF(84, 108) = 2 × 2 × 3 = 12
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