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HCF of 84 and 144

HCF of 84 and 144 is the largest possible number that divides 84 and 144 exactly without any remainder. The factors of 84 and 144 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 respectively. There are 3 commonly used methods to find the HCF of 84 and 144 - long division, prime factorization, and Euclidean algorithm.

1.	HCF of 84 and 144
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 84 and 144?

Answer: HCF of 84 and 144 is 12.

Explanation:

The HCF of two non-zero integers, x(84) and y(144), is the highest positive integer m(12) that divides both x(84) and y(144) without any remainder.

Methods to Find HCF of 84 and 144

The methods to find the HCF of 84 and 144 are explained below.

Prime Factorization Method
Long Division Method
Using Euclid's Algorithm

HCF of 84 and 144 by Prime Factorization

Prime factorization of 84 and 144 is (2 × 2 × 3 × 7) and (2 × 2 × 2 × 2 × 3 × 3) respectively. As visible, 84 and 144 have common prime factors. Hence, the HCF of 84 and 144 is 2 × 2 × 3 = 12.

HCF of 84 and 144 by Long Division

HCF of 84 and 144 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 144 (larger number) by 84 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (60).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the HCF of 84 and 144.

HCF of 84 and 144 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 = 144 and Y = 84

HCF(144, 84) = HCF(84, 144 mod 84) = HCF(84, 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 84 and 144 is 12.

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HCF of 84 and 144 Examples

Example 1: The product of two numbers is 12096. If their HCF is 12, what is their LCM?

Solution:

Given: HCF = 12 and product of numbers = 12096
∵ LCM × HCF = product of numbers
⇒ LCM = Product/HCF = 12096/12
Therefore, the LCM is 1008.
Example 2: For two numbers, HCF = 12 and LCM = 1008. If one number is 144, find the other number.

Solution:

Given: HCF (z, 144) = 12 and LCM (z, 144) = 1008
∵ HCF × LCM = 144 × (z)
⇒ z = (HCF × LCM)/144
⇒ z = (12 × 1008)/144
⇒ z = 84
Therefore, the other number is 84.
Example 3: Find the HCF of 84 and 144, if their LCM is 1008.

Solution:

∵ LCM × HCF = 84 × 144
⇒ HCF(84, 144) = (84 × 144)/1008 = 12
Therefore, the highest common factor of 84 and 144 is 12.

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FAQs on HCF of 84 and 144

What is the HCF of 84 and 144?

The HCF of 84 and 144 is 12. To calculate the HCF (Highest Common Factor) of 84 and 144, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 144 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144) and choose the highest factor that exactly divides both 84 and 144, i.e., 12.

If the HCF of 144 and 84 is 12, Find its LCM.

HCF(144, 84) × LCM(144, 84) = 144 × 84
Since the HCF of 144 and 84 = 12
⇒ 12 × LCM(144, 84) = 12096
Therefore, LCM = 1008
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What is the Relation Between LCM and HCF of 84, 144?

The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 84 and 144, i.e. HCF × LCM = 84 × 144.

How to Find the HCF of 84 and 144 by Long Division Method?

To find the HCF of 84, 144 using long division method, 144 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 144?

There are three commonly used methods to find the HCF of 84 and 144.

By Long Division
By Listing Common Factors
By Prime Factorization

How to Find the HCF of 84 and 144 by Prime Factorization?

To find the HCF of 84 and 144, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 144 = 2 × 2 × 2 × 2 × 3 × 3.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 84 and 144. Hence, HCF(84, 144) = 2 × 2 × 3 = 12
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