GCF of 84 and 96
GCF of 84 and 96 is the largest possible number that divides 84 and 96 exactly without any remainder. The factors of 84 and 96 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 respectively. There are 3 commonly used methods to find the GCF of 84 and 96  Euclidean algorithm, long division, and prime factorization.
1.  GCF of 84 and 96 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is GCF of 84 and 96?
Answer: GCF of 84 and 96 is 12.
Explanation:
The GCF of two nonzero integers, x(84) and y(96), is the greatest positive integer m(12) that divides both x(84) and y(96) without any remainder.
Methods to Find GCF of 84 and 96
Let's look at the different methods for finding the GCF of 84 and 96.
 Prime Factorization Method
 Listing Common Factors
 Using Euclid's Algorithm
GCF of 84 and 96 by Prime Factorization
Prime factorization of 84 and 96 is (2 × 2 × 3 × 7) and (2 × 2 × 2 × 2 × 2 × 3) respectively. As visible, 84 and 96 have common prime factors. Hence, the GCF of 84 and 96 is 2 × 2 × 3 = 12.
GCF of 84 and 96 by Listing Common Factors
 Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
 Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
There are 6 common factors of 84 and 96, that are 1, 2, 3, 4, 6, and 12. Therefore, the greatest common factor of 84 and 96 is 12.
GCF of 84 and 96 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 = 96 and Y = 84
 GCF(96, 84) = GCF(84, 96 mod 84) = GCF(84, 12)
 GCF(84, 12) = GCF(12, 84 mod 12) = GCF(12, 0)
 GCF(12, 0) = 12 (∵ GCF(X, 0) = X, where X ≠ 0)
Therefore, the value of GCF of 84 and 96 is 12.
☛ Also Check:
 GCF of 25 and 45 = 5
 GCF of 96 and 144 = 48
 GCF of 24 and 54 = 6
 GCF of 21 and 36 = 3
 GCF of 18 and 20 = 2
 GCF of 56 and 98 = 14
 GCF of 8 and 10 = 2
GCF of 84 and 96 Examples

Example 1: The product of two numbers is 8064. If their GCF is 12, what is their LCM?
Solution:
Given: GCF = 12 and product of numbers = 8064
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 8064/12
Therefore, the LCM is 672. 
Example 2: Find the GCF of 84 and 96, if their LCM is 672.
Solution:
∵ LCM × GCF = 84 × 96
⇒ GCF(84, 96) = (84 × 96)/672 = 12
Therefore, the greatest common factor of 84 and 96 is 12. 
Example 3: For two numbers, GCF = 12 and LCM = 672. If one number is 84, find the other number.
Solution:
Given: GCF (y, 84) = 12 and LCM (y, 84) = 672
∵ GCF × LCM = 84 × (y)
⇒ y = (GCF × LCM)/84
⇒ y = (12 × 672)/84
⇒ y = 96
Therefore, the other number is 96.
FAQs on GCF of 84 and 96
What is the GCF of 84 and 96?
The GCF of 84 and 96 is 12. To calculate the greatest common factor of 84 and 96, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96) and choose the greatest factor that exactly divides both 84 and 96, i.e., 12.
How to Find the GCF of 84 and 96 by Prime Factorization?
To find the GCF of 84 and 96, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 96 = 2 × 2 × 2 × 2 × 2 × 3.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 84 and 96. Hence, GCF(84, 96) = 2 × 2 × 3 = 12
☛ Prime Number
How to Find the GCF of 84 and 96 by Long Division Method?
To find the GCF of 84, 96 using long division method, 96 is divided by 84. The corresponding divisor (12) when remainder equals 0 is taken as GCF.
What are the Methods to Find GCF of 84 and 96?
There are three commonly used methods to find the GCF of 84 and 96.
 By Long Division
 By Prime Factorization
 By Euclidean Algorithm
If the GCF of 96 and 84 is 12, Find its LCM.
GCF(96, 84) × LCM(96, 84) = 96 × 84
Since the GCF of 96 and 84 = 12
⇒ 12 × LCM(96, 84) = 8064
Therefore, LCM = 672
☛ Greatest Common Factor Calculator
What is the Relation Between LCM and GCF of 84, 96?
The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 84 and 96, i.e. GCF × LCM = 84 × 96.
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