HCF of 8, 10 and 12
HCF of 8, 10 and 12 is the largest possible number that divides 8, 10 and 12 exactly without any remainder. The factors of 8, 10 and 12 are (1, 2, 4, 8), (1, 2, 5, 10) and (1, 2, 3, 4, 6, 12) respectively. There are 3 commonly used methods to find the HCF of 8, 10 and 12  Euclidean algorithm, prime factorization, and long division.
1.  HCF of 8, 10 and 12 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is HCF of 8, 10 and 12?
Answer: HCF of 8, 10 and 12 is 2.
Explanation:
The HCF of three nonzero integers, x(8), y(10) and z(12), is the highest positive integer m(2) that divides x(8), y(10) and z(12) without any remainder.
Methods to Find HCF of 8, 10 and 12
The methods to find the HCF of 8, 10 and 12 are explained below.
 Using Euclid's Algorithm
 Listing Common Factors
 Prime Factorization Method
HCF of 8, 10 and 12 by Euclidean Algorithm
As per the Euclidean Algorithm, HCF(X, Y) = HCF(Y, X mod Y)
where X > Y and mod is the modulo operator.
HCF(8, 10, 12) = HCF(HCF(8, 10), 12)
 HCF(10, 8) = HCF(8, 10 mod 8) = HCF(8, 2)
 HCF(8, 2) = HCF(2, 8 mod 2) = HCF(2, 0)
 HCF(2, 0) = 2 (∵ HCF(X, 0) = X, where X ≠ 0)
Steps for HCF(2, 12)
 HCF(12, 2) = HCF(2, 12 mod 2) = HCF(2, 0)
 HCF(2, 0) = 2 (∵ HCF(X, 0) = X, where X ≠ 0)
Therefore, the value of HCF of 8, 10 and 12 is 2.
HCF of 8, 10 and 12 by Listing Common Factors
 Factors of 8: 1, 2, 4, 8
 Factors of 10: 1, 2, 5, 10
 Factors of 12: 1, 2, 3, 4, 6, 12
There are 2 common factors of 8, 10 and 12, that are 1 and 2. Therefore, the highest common factor of 8, 10 and 12 is 2.
HCF of 8, 10 and 12 by Prime Factorization
Prime factorization of 8, 10 and 12 is (2 × 2 × 2), (2 × 5) and (2 × 2 × 3) respectively. As visible, 8, 10 and 12 have only one common prime factor i.e. 2. Hence, the HCF of 8, 10 and 12 is 2.
☛ Also Check:
 HCF of 726 and 275 = 11
 HCF of 336 and 54 = 6
 HCF of 150 and 225 = 75
 HCF of 25 and 36 = 1
 HCF of 120, 144 and 204 = 12
 HCF of 408 and 1032 = 24
 HCF of 12 and 24 = 12
HCF of 8, 10 and 12 Examples

Example 1: Calculate the HCF of 8, 10, and 12 using LCM of the given numbers.
Solution:
Prime factorization of 8, 10 and 12 is given as,
 8 = 2 × 2 × 2
 10 = 2 × 5
 12 = 2 × 2 × 3
LCM(8, 10) = 40, LCM(10, 12) = 60, LCM(12, 8) = 24, LCM(8, 10, 12) = 120
⇒ HCF(8, 10, 12) = [(8 × 10 × 12) × LCM(8, 10, 12)]/[LCM(8, 10) × LCM (10, 12) × LCM(12, 8)]
⇒ HCF(8, 10, 12) = (960 × 120)/(40 × 60 × 24)
⇒ HCF(8, 10, 12) = 2.
Therefore, the HCF of 8, 10 and 12 is 2. 
Example 2: Find the highest number that divides 8, 10, and 12 completely.
Solution:
The highest number that divides 8, 10, and 12 exactly is their highest common factor.
 Factors of 8 = 1, 2, 4, 8
 Factors of 10 = 1, 2, 5, 10
 Factors of 12 = 1, 2, 3, 4, 6, 12
The HCF of 8, 10, and 12 is 2.
∴ The highest number that divides 8, 10, and 12 is 2. 
Example 3: Verify the relation between the LCM and HCF of 8, 10 and 12.
Solution:
The relation between the LCM and HCF of 8, 10 and 12 is given as, HCF(8, 10, 12) = [(8 × 10 × 12) × LCM(8, 10, 12)]/[LCM(8, 10) × LCM (10, 12) × LCM(8, 12)]
⇒ Prime factorization of 8, 10 and 12: 8 = 2 × 2 × 2
 10 = 2 × 5
 12 = 2 × 2 × 3
∴ LCM of (8, 10), (10, 12), (8, 12), and (8, 10, 12) is 40, 60, 24, and 120 respectively.
Now, LHS = HCF(8, 10, 12) = 2.
And, RHS = [(8 × 10 × 12) × LCM(8, 10, 12)]/[LCM(8, 10) × LCM (10, 12) × LCM(8, 12)] = [(960) × 120]/[40 × 60 × 24]
LHS = RHS = 2.
Hence verified.
FAQs on HCF of 8, 10 and 12
What is the HCF of 8, 10 and 12?
The HCF of 8, 10 and 12 is 2. To calculate the HCF of 8, 10 and 12, we need to factor each number (factors of 8 = 1, 2, 4, 8; factors of 10 = 1, 2, 5, 10; factors of 12 = 1, 2, 3, 4, 6, 12) and choose the highest factor that exactly divides 8, 10 and 12, i.e., 2.
Which of the following is HCF of 8, 10 and 12? 2, 31, 46, 56, 17
HCF of 8, 10, 12 will be the number that divides 8, 10, and 12 without leaving any remainder. The only number that satisfies the given condition is 2.
How to Find the HCF of 8, 10 and 12 by Prime Factorization?
To find the HCF of 8, 10 and 12, we will find the prime factorization of given numbers, i.e. 8 = 2 × 2 × 2; 10 = 2 × 5; 12 = 2 × 2 × 3.
⇒ Since 2 is the only common prime factor of 8, 10 and 12. Hence, HCF(8, 10, 12) = 2.
☛ Prime Number
What is the Relation Between LCM and HCF of 8, 10 and 12?
The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 8, 10 and 12, i.e. HCF(8, 10, 12) = [(8 × 10 × 12) × LCM(8, 10, 12)]/[LCM(8, 10) × LCM (10, 12) × LCM(8, 12)].
☛ Highest Common Factor Calculator
What are the Methods to Find HCF of 8, 10 and 12?
There are three commonly used methods to find the HCF of 8, 10 and 12.
 By Long Division
 By Euclidean Algorithm
 By Prime Factorization
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