GCF of 84 and 126
GCF of 84 and 126 is the largest possible number that divides 84 and 126 exactly without any remainder. The factors of 84 and 126 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126 respectively. There are 3 commonly used methods to find the GCF of 84 and 126  Euclidean algorithm, prime factorization, and long division.
1.  GCF of 84 and 126 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is GCF of 84 and 126?
Answer: GCF of 84 and 126 is 42.
Explanation:
The GCF of two nonzero integers, x(84) and y(126), is the greatest positive integer m(42) that divides both x(84) and y(126) without any remainder.
Methods to Find GCF of 84 and 126
The methods to find the GCF of 84 and 126 are explained below.
 Prime Factorization Method
 Using Euclid's Algorithm
 Long Division Method
GCF of 84 and 126 by Prime Factorization
Prime factorization of 84 and 126 is (2 × 2 × 3 × 7) and (2 × 3 × 3 × 7) respectively. As visible, 84 and 126 have common prime factors. Hence, the GCF of 84 and 126 is 2 × 3 × 7 = 42.
GCF of 84 and 126 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 = 126 and Y = 84
 GCF(126, 84) = GCF(84, 126 mod 84) = GCF(84, 42)
 GCF(84, 42) = GCF(42, 84 mod 42) = GCF(42, 0)
 GCF(42, 0) = 42 (∵ GCF(X, 0) = X, where X ≠ 0)
Therefore, the value of GCF of 84 and 126 is 42.
GCF of 84 and 126 by Long Division
GCF of 84 and 126 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.
 Step 1: Divide 126 (larger number) by 84 (smaller number).
 Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (42).
 Step 3: Repeat this process until the remainder = 0.
The corresponding divisor (42) is the GCF of 84 and 126.
☛ Also Check:
 GCF of 5 and 8 = 1
 GCF of 20 and 45 = 5
 GCF of 22 and 33 = 11
 GCF of 51 and 85 = 17
 GCF of 27 and 72 = 9
 GCF of 10 and 50 = 10
 GCF of 32 and 36 = 4
GCF of 84 and 126 Examples

Example 1: The product of two numbers is 10584. If their GCF is 42, what is their LCM?
Solution:
Given: GCF = 42 and product of numbers = 10584
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 10584/42
Therefore, the LCM is 252. 
Example 2: Find the GCF of 84 and 126, if their LCM is 252.
Solution:
∵ LCM × GCF = 84 × 126
⇒ GCF(84, 126) = (84 × 126)/252 = 42
Therefore, the greatest common factor of 84 and 126 is 42. 
Example 3: For two numbers, GCF = 42 and LCM = 252. If one number is 84, find the other number.
Solution:
Given: GCF (y, 84) = 42 and LCM (y, 84) = 252
∵ GCF × LCM = 84 × (y)
⇒ y = (GCF × LCM)/84
⇒ y = (42 × 252)/84
⇒ y = 126
Therefore, the other number is 126.
FAQs on GCF of 84 and 126
What is the GCF of 84 and 126?
The GCF of 84 and 126 is 42. To calculate the greatest common factor of 84 and 126, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 126 = 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126) and choose the greatest factor that exactly divides both 84 and 126, i.e., 42.
If the GCF of 126 and 84 is 42, Find its LCM.
GCF(126, 84) × LCM(126, 84) = 126 × 84
Since the GCF of 126 and 84 = 42
⇒ 42 × LCM(126, 84) = 10584
Therefore, LCM = 252
☛ Greatest Common Factor Calculator
How to Find the GCF of 84 and 126 by Prime Factorization?
To find the GCF of 84 and 126, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 126 = 2 × 3 × 3 × 7.
⇒ Since 2, 3, 7 are common terms in the prime factorization of 84 and 126. Hence, GCF(84, 126) = 2 × 3 × 7 = 42
☛ What are Prime Numbers?
How to Find the GCF of 84 and 126 by Long Division Method?
To find the GCF of 84, 126 using long division method, 126 is divided by 84. The corresponding divisor (42) when remainder equals 0 is taken as GCF.
What is the Relation Between LCM and GCF of 84, 126?
The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 84 and 126, i.e. GCF × LCM = 84 × 126.
What are the Methods to Find GCF of 84 and 126?
There are three commonly used methods to find the GCF of 84 and 126.
 By Listing Common Factors
 By Prime Factorization
 By Long Division
visual curriculum