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HCF of 60 and 84

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

1.	HCF of 60 and 84
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 60 and 84?

Answer: HCF of 60 and 84 is 12.

Explanation:

The HCF of two non-zero integers, x(60) and y(84), is the highest positive integer m(12) that divides both x(60) and y(84) without any remainder.

Methods to Find HCF of 60 and 84

The methods to find the HCF of 60 and 84 are explained below.

Listing Common Factors
Long Division Method
Using Euclid's Algorithm

HCF of 60 and 84 by Listing Common Factors

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

There are 6 common factors of 60 and 84, that are 1, 2, 3, 4, 6, and 12. Therefore, the highest common factor of 60 and 84 is 12.

HCF of 60 and 84 by Long Division

HCF of 60 and 84 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 84 (larger number) by 60 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (60) by the remainder (24).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the HCF of 60 and 84.

HCF of 60 and 84 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 = 84 and Y = 60

HCF(84, 60) = HCF(60, 84 mod 60) = HCF(60, 24)
HCF(60, 24) = HCF(24, 60 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 60 and 84 is 12.

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HCF of 60 and 84 Examples

Example 1: The product of two numbers is 5040. If their HCF is 12, what is their LCM?

Solution:

Given: HCF = 12 and product of numbers = 5040
∵ LCM × HCF = product of numbers
⇒ LCM = Product/HCF = 5040/12
Therefore, the LCM is 420.
Example 2: For two numbers, HCF = 12 and LCM = 420. If one number is 60, find the other number.

Solution:

Given: HCF (z, 60) = 12 and LCM (z, 60) = 420
∵ HCF × LCM = 60 × (z)
⇒ z = (HCF × LCM)/60
⇒ z = (12 × 420)/60
⇒ z = 84
Therefore, the other number is 84.
Example 3: Find the HCF of 60 and 84, if their LCM is 420.

Solution:

∵ LCM × HCF = 60 × 84
⇒ HCF(60, 84) = (60 × 84)/420 = 12
Therefore, the highest common factor of 60 and 84 is 12.

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FAQs on HCF of 60 and 84

What is the HCF of 60 and 84?

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

If the HCF of 84 and 60 is 12, Find its LCM.

HCF(84, 60) × LCM(84, 60) = 84 × 60
Since the HCF of 84 and 60 = 12
⇒ 12 × LCM(84, 60) = 5040
Therefore, LCM = 420
☛ Highest Common Factor Calculator

What is the Relation Between LCM and HCF of 60, 84?

The following equation can be used to express the relation between LCM and HCF of 60 and 84, i.e. HCF × LCM = 60 × 84.

What are the Methods to Find HCF of 60 and 84?

There are three commonly used methods to find the HCF of 60 and 84.

By Long Division
By Listing Common Factors
By Prime Factorization

How to Find the HCF of 60 and 84 by Prime Factorization?

To find the HCF of 60 and 84, we will find the prime factorization of the given numbers, i.e. 60 = 2 × 2 × 3 × 5; 84 = 2 × 2 × 3 × 7.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 60 and 84. Hence, HCF(60, 84) = 2 × 2 × 3 = 12
☛ Prime Numbers

How to Find the HCF of 60 and 84 by Long Division Method?

To find the HCF of 60, 84 using long division method, 84 is divided by 60. The corresponding divisor (12) when remainder equals 0 is taken as HCF.

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