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HCF of 84 and 120

HCF of 84 and 120 is the largest possible number that divides 84 and 120 exactly without any remainder. The factors of 84 and 120 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 respectively. There are 3 commonly used methods to find the HCF of 84 and 120 - Euclidean algorithm, prime factorization, and long division.

1.	HCF of 84 and 120
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 84 and 120?

Answer: HCF of 84 and 120 is 12.

Explanation:

The HCF of two non-zero integers, x(84) and y(120), is the highest positive integer m(12) that divides both x(84) and y(120) without any remainder.

Methods to Find HCF of 84 and 120

Let's look at the different methods for finding the HCF of 84 and 120.

Listing Common Factors
Long Division Method
Using Euclid's Algorithm

HCF of 84 and 120 by Listing Common Factors

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

There are 6 common factors of 84 and 120, that are 1, 2, 3, 4, 6, and 12. Therefore, the highest common factor of 84 and 120 is 12.

HCF of 84 and 120 by Long Division

HCF of 84 and 120 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 120 (larger number) by 84 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (36).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the HCF of 84 and 120.

HCF of 84 and 120 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 = 120 and Y = 84

HCF(120, 84) = HCF(84, 120 mod 84) = HCF(84, 36)
HCF(84, 36) = HCF(36, 84 mod 36) = HCF(36, 12)
HCF(36, 12) = HCF(12, 36 mod 12) = HCF(12, 0)
HCF(12, 0) = 12 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 84 and 120 is 12.

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HCF of 84 and 120 Examples

Example 1: For two numbers, HCF = 12 and LCM = 840. If one number is 84, find the other number.

Solution:

Given: HCF (z, 84) = 12 and LCM (z, 84) = 840
∵ HCF × LCM = 84 × (z)
⇒ z = (HCF × LCM)/84
⇒ z = (12 × 840)/84
⇒ z = 120
Therefore, the other number is 120.
Example 2: Find the HCF of 84 and 120, if their LCM is 840.

Solution:

∵ LCM × HCF = 84 × 120
⇒ HCF(84, 120) = (84 × 120)/840 = 12
Therefore, the highest common factor of 84 and 120 is 12.
Example 3: The product of two numbers is 10080. If their HCF is 12, what is their LCM?

Solution:

Given: HCF = 12 and product of numbers = 10080
∵ LCM × HCF = product of numbers
⇒ LCM = Product/HCF = 10080/12
Therefore, the LCM is 840.

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FAQs on HCF of 84 and 120

What is the HCF of 84 and 120?

The HCF of 84 and 120 is 12. To calculate the HCF (Highest Common Factor) of 84 and 120, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120) and choose the highest factor that exactly divides both 84 and 120, i.e., 12.

How to Find the HCF of 84 and 120 by Long Division Method?

To find the HCF of 84, 120 using long division method, 120 is divided by 84. The corresponding divisor (12) when remainder equals 0 is taken as HCF.

What are the Methods to Find HCF of 84 and 120?

There are three commonly used methods to find the HCF of 84 and 120.

By Prime Factorization
By Euclidean Algorithm
By Long Division

If the HCF of 120 and 84 is 12, Find its LCM.

HCF(120, 84) × LCM(120, 84) = 120 × 84
Since the HCF of 120 and 84 = 12
⇒ 12 × LCM(120, 84) = 10080
Therefore, LCM = 840
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What is the Relation Between LCM and HCF of 84, 120?

The following equation can be used to express the relation between LCM (Least Common Multiple) and HCF of 84 and 120, i.e. HCF × LCM = 84 × 120.

How to Find the HCF of 84 and 120 by Prime Factorization?

To find the HCF of 84 and 120, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 120 = 2 × 2 × 2 × 3 × 5.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 84 and 120. Hence, HCF(84, 120) = 2 × 2 × 3 = 12
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