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HCF of 84 and 108

HCF of 84 and 108 is the largest possible number that divides 84 and 108 exactly without any remainder. The factors of 84 and 108 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 respectively. There are 3 commonly used methods to find the HCF of 84 and 108 - Euclidean algorithm, long division, and prime factorization.

1.	HCF of 84 and 108
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 84 and 108?

Answer: HCF of 84 and 108 is 12.

Explanation:

The HCF of two non-zero integers, x(84) and y(108), is the highest positive integer m(12) that divides both x(84) and y(108) without any remainder.

Methods to Find HCF of 84 and 108

The methods to find the HCF of 84 and 108 are explained below.

Using Euclid's Algorithm
Prime Factorization Method
Long Division Method

HCF of 84 and 108 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.

Here X = 108 and Y = 84

HCF(108, 84) = HCF(84, 108 mod 84) = HCF(84, 24)
HCF(84, 24) = HCF(24, 84 mod 24) = HCF(24, 12)
HCF(24, 12) = HCF(12, 24 mod 12) = HCF(12, 0)
HCF(12, 0) = 12 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 84 and 108 is 12.

HCF of 84 and 108 by Prime Factorization

Prime factorization of 84 and 108 is (2 × 2 × 3 × 7) and (2 × 2 × 3 × 3 × 3) respectively. As visible, 84 and 108 have common prime factors. Hence, the HCF of 84 and 108 is 2 × 2 × 3 = 12.

HCF of 84 and 108 by Long Division

HCF of 84 and 108 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 108 (larger number) by 84 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (24).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the HCF of 84 and 108.

☛ Also Check:

HCF of 84 and 108 Examples

Example 1: Find the HCF of 84 and 108, if their LCM is 756.

Solution:

∵ LCM × HCF = 84 × 108
⇒ HCF(84, 108) = (84 × 108)/756 = 12
Therefore, the highest common factor of 84 and 108 is 12.
Example 2: The product of two numbers is 9072. If their HCF is 12, what is their LCM?

Solution:

Given: HCF = 12 and product of numbers = 9072
∵ LCM × HCF = product of numbers
⇒ LCM = Product/HCF = 9072/12
Therefore, the LCM is 756.
Example 3: Find the highest number that divides 84 and 108 exactly.

Solution:

The highest number that divides 84 and 108 exactly is their highest common factor, i.e. HCF of 84 and 108.
⇒ Factors of 84 and 108:
- Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Factors of 108 = 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Therefore, the HCF of 84 and 108 is 12.

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FAQs on HCF of 84 and 108

What is the HCF of 84 and 108?

The HCF of 84 and 108 is 12. To calculate the HCF (Highest Common Factor) of 84 and 108, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 108 = 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108) and choose the highest factor that exactly divides both 84 and 108, i.e., 12.

How to Find the HCF of 84 and 108 by Prime Factorization?

To find the HCF of 84 and 108, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 108 = 2 × 2 × 3 × 3 × 3.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 84 and 108. Hence, HCF(84, 108) = 2 × 2 × 3 = 12
☛ Prime Numbers

If the HCF of 108 and 84 is 12, Find its LCM.

HCF(108, 84) × LCM(108, 84) = 108 × 84
Since the HCF of 108 and 84 = 12
⇒ 12 × LCM(108, 84) = 9072
Therefore, LCM = 756
☛ Highest Common Factor Calculator

How to Find the HCF of 84 and 108 by Long Division Method?

To find the HCF of 84, 108 using long division method, 108 is divided by 84. The corresponding divisor (12) when remainder equals 0 is taken as HCF.

What is the Relation Between LCM and HCF of 84, 108?

The following equation can be used to express the relation between Least Common Multiple and HCF of 84 and 108, i.e. HCF × LCM = 84 × 108.

What are the Methods to Find HCF of 84 and 108?

There are three commonly used methods to find the HCF of 84 and 108.

By Long Division
By Listing Common Factors
By Prime Factorization

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