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GCF of 42, 70 and 84
GCF of 42, 70 and 84 is the largest possible number that divides 42, 70 and 84 exactly without any remainder. The factors of 42, 70 and 84 are (1, 2, 3, 6, 7, 14, 21, 42), (1, 2, 5, 7, 10, 14, 35, 70) and (1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) respectively. There are 3 commonly used methods to find the GCF of 42, 70 and 84  long division, Euclidean algorithm, and prime factorization.
1.  GCF of 42, 70 and 84 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is GCF of 42, 70 and 84?
Answer: GCF of 42, 70 and 84 is 14.
Explanation:
The GCF of three nonzero integers, x(42), y(70) and z(84), is the greatest positive integer m(14) that divides x(42), y(70) and z(84) without any remainder.
Methods to Find GCF of 42, 70 and 84
The methods to find the GCF of 42, 70 and 84 are explained below.
 Listing Common Factors
 Long Division Method
 Prime Factorization Method
GCF of 42, 70 and 84 by Listing Common Factors
 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
 Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
 Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
There are 4 common factors of 42, 70 and 84, that are 1, 2, 14, and 7. Therefore, the greatest common factor of 42, 70 and 84 is 14.
GCF of 42, 70 and 84 by Long Division
GCF of 42, 70 and 84 can be represented as GCF of (GCF of 42, 70) and 84. GCF(42, 70, 84) can be thus calculated by first finding GCF(42, 70) using long division and thereafter using this result with 84 to perform long division again.
 Step 1: Divide 70 (larger number) by 42 (smaller number).
 Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (42) by the remainder (28). Repeat this process until the remainder = 0.
⇒ GCF(42, 70) = 14.  Step 3: Now to find the GCF of 14 and 84, we will perform a long division on 84 and 14.
 Step 4: For remainder = 0, divisor = 14 ⇒ GCF(14, 84) = 14
Thus, GCF(42, 70, 84) = GCF(GCF(42, 70), 84) = 14.
GCF of 42, 70 and 84 by Prime Factorization
Prime factorization of 42, 70 and 84 is (2 × 3 × 7), (2 × 5 × 7) and (2 × 2 × 3 × 7) respectively. As visible, 42, 70 and 84 have common prime factors. Hence, the GCF of 42, 70 and 84 is 2 × 7 = 14.
☛ Also Check:
 GCF of 20 and 24 = 4
 GCF of 28 and 42 = 14
 GCF of 24 and 60 = 12
 GCF of 40 and 80 = 40
 GCF of 45 and 90 = 45
 GCF of 54 and 32 = 2
 GCF of 45 and 120 = 15
GCF of 42, 70 and 84 Examples

Example 1: Verify the relation between the LCM and GCF of 42, 70 and 84.
Solution:
The relation between the LCM and GCF of 42, 70 and 84 is given as, GCF(42, 70, 84) = [(42 × 70 × 84) × LCM(42, 70, 84)]/[LCM(42, 70) × LCM (70, 84) × LCM(42, 84)]
⇒ Prime factorization of 42, 70 and 84: 42 = 2 × 3 × 7
 70 = 2 × 5 × 7
 84 = 2 × 2 × 3 × 7
∴ LCM of (42, 70), (70, 84), (42, 84), and (42, 70, 84) is 210, 420, 84, and 420 respectively.
Now, LHS = GCF(42, 70, 84) = 14.
And, RHS = [(42 × 70 × 84) × LCM(42, 70, 84)]/[LCM(42, 70) × LCM (70, 84) × LCM(42, 84)] = [(246960) × 420]/[210 × 420 × 84]
LHS = RHS = 14.
Hence verified. 
Example 2: Calculate the GCF of 42, 70, and 84 using LCM of the given numbers.
Solution:
Prime factorization of 42, 70 and 84 is given as,
 42 = 2 × 3 × 7
 70 = 2 × 5 × 7
 84 = 2 × 2 × 3 × 7
LCM(42, 70) = 210, LCM(70, 84) = 420, LCM(84, 42) = 84, LCM(42, 70, 84) = 420
⇒ GCF(42, 70, 84) = [(42 × 70 × 84) × LCM(42, 70, 84)]/[LCM(42, 70) × LCM (70, 84) × LCM(84, 42)]
⇒ GCF(42, 70, 84) = (246960 × 420)/(210 × 420 × 84)
⇒ GCF(42, 70, 84) = 14.
Therefore, the GCF of 42, 70 and 84 is 14. 
Example 3: Find the greatest number that divides 42, 70, and 84 completely.
Solution:
The greatest number that divides 42, 70, and 84 exactly is their greatest common factor.
 Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42
 Factors of 70 = 1, 2, 5, 7, 10, 14, 35, 70
 Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
The GCF of 42, 70, and 84 is 14.
∴ The greatest number that divides 42, 70, and 84 is 14.
FAQs on GCF of 42, 70 and 84
What is the GCF of 42, 70 and 84?
The GCF of 42, 70 and 84 is 14. To calculate the greatest common factor of 42, 70 and 84, we need to factor each number (factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42; factors of 70 = 1, 2, 5, 7, 10, 14, 35, 70; factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) and choose the greatest factor that exactly divides 42, 70 and 84, i.e., 14.
Which of the following is GCF of 42, 70 and 84? 14, 89, 118, 100, 96, 101, 88, 123
GCF of 42, 70, 84 will be the number that divides 42, 70, and 84 without leaving any remainder. The only number that satisfies the given condition is 14.
What is the Relation Between LCM and GCF of 42, 70 and 84?
The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 42, 70 and 84, i.e. GCF(42, 70, 84) = [(42 × 70 × 84) × LCM(42, 70, 84)]/[LCM(42, 70) × LCM (70, 84) × LCM(42, 84)].
☛ GCF Calculator
What are the Methods to Find GCF of 42, 70 and 84?
There are three commonly used methods to find the GCF of 42, 70 and 84.
 By Prime Factorization
 By Long Division
 By Euclidean Algorithm
How to Find the GCF of 42, 70 and 84 by Prime Factorization?
To find the GCF of 42, 70 and 84, we will find the prime factorization of given numbers, i.e. 42 = 2 × 3 × 7; 70 = 2 × 5 × 7; 84 = 2 × 2 × 3 × 7.
⇒ Since 2, 7 are common terms in the prime factorization of 42, 70 and 84. Hence, GCF(42, 70, 84) = 2 × 7 = 14
☛ Prime Number
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