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HCF of 120 and 168

HCF of 120 and 168 is the largest possible number that divides 120 and 168 exactly without any remainder. The factors of 120 and 168 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 and 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168 respectively. There are 3 commonly used methods to find the HCF of 120 and 168 - Euclidean algorithm, long division, and prime factorization.

1.	HCF of 120 and 168
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 120 and 168?

Answer: HCF of 120 and 168 is 24.

Explanation:

The HCF of two non-zero integers, x(120) and y(168), is the highest positive integer m(24) that divides both x(120) and y(168) without any remainder.

Methods to Find HCF of 120 and 168

Let's look at the different methods for finding the HCF of 120 and 168.

Listing Common Factors
Long Division Method
Using Euclid's Algorithm

HCF of 120 and 168 by Listing Common Factors

Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168

There are 8 common factors of 120 and 168, that are 1, 2, 3, 4, 6, 8, 12, and 24. Therefore, the highest common factor of 120 and 168 is 24.

HCF of 120 and 168 by Long Division

HCF of 120 and 168 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 168 (larger number) by 120 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (120) by the remainder (48).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (24) is the HCF of 120 and 168.

HCF of 120 and 168 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 = 168 and Y = 120

HCF(168, 120) = HCF(120, 168 mod 120) = HCF(120, 48)
HCF(120, 48) = HCF(48, 120 mod 48) = HCF(48, 24)
HCF(48, 24) = HCF(24, 48 mod 24) = HCF(24, 0)
HCF(24, 0) = 24 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 120 and 168 is 24.

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HCF of 120 and 168 Examples

Example 1: The product of two numbers is 20160. If their HCF is 24, what is their LCM?

Solution:

Given: HCF = 24 and product of numbers = 20160
∵ LCM × HCF = product of numbers
⇒ LCM = Product/HCF = 20160/24
Therefore, the LCM is 840.
Example 2: Find the HCF of 120 and 168, if their LCM is 840.

Solution:

∵ LCM × HCF = 120 × 168
⇒ HCF(120, 168) = (120 × 168)/840 = 24
Therefore, the highest common factor of 120 and 168 is 24.
Example 3: For two numbers, HCF = 24 and LCM = 840. If one number is 168, find the other number.

Solution:

Given: HCF (y, 168) = 24 and LCM (y, 168) = 840
∵ HCF × LCM = 168 × (y)
⇒ y = (HCF × LCM)/168
⇒ y = (24 × 840)/168
⇒ y = 120
Therefore, the other number is 120.

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FAQs on HCF of 120 and 168

What is the HCF of 120 and 168?

The HCF of 120 and 168 is 24. To calculate the HCF (Highest Common Factor) of 120 and 168, we need to factor each number (factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120; factors of 168 = 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168) and choose the highest factor that exactly divides both 120 and 168, i.e., 24.

How to Find the HCF of 120 and 168 by Long Division Method?

To find the HCF of 120, 168 using long division method, 168 is divided by 120. The corresponding divisor (24) when remainder equals 0 is taken as HCF.

How to Find the HCF of 120 and 168 by Prime Factorization?

To find the HCF of 120 and 168, we will find the prime factorization of the given numbers, i.e. 120 = 2 × 2 × 2 × 3 × 5; 168 = 2 × 2 × 2 × 3 × 7.
⇒ Since 2, 2, 2, 3 are common terms in the prime factorization of 120 and 168. Hence, HCF(120, 168) = 2 × 2 × 2 × 3 = 24
☛ What is a Prime Number?

What is the Relation Between LCM and HCF of 120, 168?

The following equation can be used to express the relation between Least Common Multiple and HCF of 120 and 168, i.e. HCF × LCM = 120 × 168.

If the HCF of 168 and 120 is 24, Find its LCM.

HCF(168, 120) × LCM(168, 120) = 168 × 120
Since the HCF of 168 and 120 = 24
⇒ 24 × LCM(168, 120) = 20160
Therefore, LCM = 840
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What are the Methods to Find HCF of 120 and 168?

There are three commonly used methods to find the HCF of 120 and 168.

By Long Division
By Prime Factorization
By Listing Common Factors

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