GCF of 75, 8 and 21
GCF of 75, 8 and 21 is the largest possible number that divides 75, 8 and 21 exactly without any remainder. The factors of 75, 8 and 21 are (1, 3, 5, 15, 25, 75), (1, 2, 4, 8) and (1, 3, 7, 21) respectively. There are 3 commonly used methods to find the GCF of 75, 8 and 21  Euclidean algorithm, prime factorization, and long division.
1.  GCF of 75, 8 and 21 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is GCF of 75, 8 and 21?
Answer: GCF of 75, 8 and 21 is 1.
Explanation:
The GCF of three nonzero integers, x(75), y(8) and z(21), is the greatest positive integer m(1) that divides x(75), y(8) and z(21) without any remainder.
Methods to Find GCF of 75, 8 and 21
Let's look at the different methods for finding the GCF of 75, 8 and 21.
 Listing Common Factors
 Prime Factorization Method
 Using Euclid's Algorithm
GCF of 75, 8 and 21 by Listing Common Factors
 Factors of 75: 1, 3, 5, 15, 25, 75
 Factors of 8: 1, 2, 4, 8
 Factors of 21: 1, 3, 7, 21
Since, 1 is the only common factor between 75, 8 and 21. The Greatest Common Factor of 75, 8 and 21 is 1.
GCF of 75, 8 and 21 by Prime Factorization
Prime factorization of 75, 8 and 21 is (3 × 5 × 5), (2 × 2 × 2) and (3 × 7) respectively. As visible, there are no common prime factors between 75, 8 and 21, i.e. they are coprime. Hence, the GCF of 75, 8 and 21 will be 1.
GCF of 75, 8 and 21 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.
GCF(75, 8, 21) = GCF(GCF(75, 8), 21)
 GCF(75, 8) = GCF(8, 75 mod 8) = GCF(8, 3)
 GCF(8, 3) = GCF(3, 8 mod 3) = GCF(3, 2)
 GCF(3, 2) = GCF(2, 3 mod 2) = GCF(2, 1)
 GCF(2, 1) = 1 (∵ GCF(X, 1) = 1)
Steps for GCF(1, 21)
 GCF(21, 1) = GCF(1, 21 mod 1) = GCF(1, 0)
 GCF(1, 0) = 1 (∵ GCF(X, 0) = X, where X ≠ 0)
Therefore, the value of GCF of 75, 8 and 21 is 1.
☛ Also Check:
 GCF of 14 and 20 = 2
 GCF of 81 and 108 = 27
 GCF of 40 and 50 = 10
 GCF of 3 and 12 = 3
 GCF of 42 and 60 = 6
 GCF of 7 and 35 = 7
 GCF of 15 and 40 = 5
GCF of 75, 8 and 21 Examples

Example 1: Find the greatest number that divides 75, 8, and 21 completely.
Solution:
The greatest number that divides 75, 8, and 21 exactly is their greatest common factor.
 Factors of 75 = 1, 3, 5, 15, 25, 75
 Factors of 8 = 1, 2, 4, 8
 Factors of 21 = 1, 3, 7, 21
The GCF of 75, 8, and 21 is 1.
∴ The greatest number that divides 75, 8, and 21 is 1. 
Example 2: Calculate the GCF of 75, 8, and 21 using LCM of the given numbers.
Solution:
Prime factorization of 75, 8 and 21 is given as,
 75 = 3 × 5 × 5
 8 = 2 × 2 × 2
 21 = 3 × 7
LCM(75, 8) = 600, LCM(8, 21) = 168, LCM(21, 75) = 525, LCM(75, 8, 21) = 4200
⇒ GCF(75, 8, 21) = [(75 × 8 × 21) × LCM(75, 8, 21)]/[LCM(75, 8) × LCM (8, 21) × LCM(21, 75)]
⇒ GCF(75, 8, 21) = (12600 × 4200)/(600 × 168 × 525)
⇒ GCF(75, 8, 21) = 1.
Therefore, the GCF of 75, 8 and 21 is 1. 
Example 3: Verify the relation between the LCM and GCF of 75, 8 and 21.
Solution:
The relation between the LCM and GCF of 75, 8 and 21 is given as, GCF(75, 8, 21) = [(75 × 8 × 21) × LCM(75, 8, 21)]/[LCM(75, 8) × LCM (8, 21) × LCM(75, 21)]
⇒ Prime factorization of 75, 8 and 21: 75 = 3 × 5 × 5
 8 = 2 × 2 × 2
 21 = 3 × 7
∴ LCM of (75, 8), (8, 21), (75, 21), and (75, 8, 21) is 600, 168, 525, and 4200 respectively.
Now, LHS = GCF(75, 8, 21) = 1.
And, RHS = [(75 × 8 × 21) × LCM(75, 8, 21)]/[LCM(75, 8) × LCM (8, 21) × LCM(75, 21)] = [(12600) × 4200]/[600 × 168 × 525]
LHS = RHS = 1.
Hence verified.
FAQs on GCF of 75, 8 and 21
What is the GCF of 75, 8 and 21?
The GCF of 75, 8 and 21 is 1. To calculate the GCF (Greatest Common Factor) of 75, 8 and 21, we need to factor each number (factors of 75 = 1, 3, 5, 15, 25, 75; factors of 8 = 1, 2, 4, 8; factors of 21 = 1, 3, 7, 21) and choose the greatest factor that exactly divides 75, 8 and 21, i.e., 1.
Which of the following is GCF of 75, 8 and 21? 1, 101, 111, 101, 114, 95, 86, 91
GCF of 75, 8, 21 will be the number that divides 75, 8, and 21 without leaving any remainder. The only number that satisfies the given condition is 1.
How to Find the GCF of 75, 8 and 21 by Prime Factorization?
To find the GCF of 75, 8 and 21, we will find the prime factorization of given numbers, i.e. 75 = 3 × 5 × 5; 8 = 2 × 2 × 2; 21 = 3 × 7.
⇒ There is no common prime factor for 75, 8 and 21. Hence, GCF(75, 8, 21) = 1.
☛ What are Prime Numbers?
What are the Methods to Find GCF of 75, 8 and 21?
There are three commonly used methods to find the GCF of 75, 8 and 21.
 By Euclidean Algorithm
 By Prime Factorization
 By Long Division
What is the Relation Between LCM and GCF of 75, 8 and 21?
The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 75, 8 and 21, i.e. GCF(75, 8, 21) = [(75 × 8 × 21) × LCM(75, 8, 21)]/[LCM(75, 8) × LCM (8, 21) × LCM(75, 21)].
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