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HCF of 6, 72 and 120

HCF of 6, 72 and 120 is the largest possible number that divides 6, 72 and 120 exactly without any remainder. The factors of 6, 72 and 120 are (1, 2, 3, 6), (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) 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 6, 72 and 120 - prime factorization, Euclidean algorithm, and long division.

1.	HCF of 6, 72 and 120
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 6, 72 and 120?

Answer: HCF of 6, 72 and 120 is 6.

Explanation:

The HCF of three non-zero integers, x(6), y(72) and z(120), is the highest positive integer m(6) that divides x(6), y(72) and z(120) without any remainder.

Methods to Find HCF of 6, 72 and 120

Let's look at the different methods for finding the HCF of 6, 72 and 120.

Prime Factorization Method
Long Division Method
Using Euclid's Algorithm

HCF of 6, 72 and 120 by Prime Factorization

Prime factorization of 6, 72 and 120 is (2 × 3), (2 × 2 × 2 × 3 × 3) and (2 × 2 × 2 × 3 × 5) respectively. As visible, 6, 72 and 120 have common prime factors. Hence, the HCF of 6, 72 and 120 is 2 × 3 = 6.

HCF of 6, 72 and 120 by Long Division

HCF of 6, 72 and 120 can be represented as HCF of (HCF of 6, 72) and 120. HCF(6, 72, 120) can be thus calculated by first finding HCF(6, 72) using long division and thereafter using this result with 120 to perform long division again.

Step 1: Divide 72 (larger number) by 6 (smaller number).
Step 2: Since the remainder = 0, the divisor (6) is the HCF(6, 72) = 6.
Step 3: Now to find the HCF of 6 and 120, we will perform a long division on 120 and 6.
Step 4: For remainder = 0, divisor = 6 ⇒ HCF(6, 120) = 6

Thus, HCF(6, 72, 120) = HCF(HCF(6, 72), 120) = 6.

HCF of 6, 72 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.

HCF(6, 72, 120) = HCF(HCF(6, 72), 120)

HCF(72, 6) = HCF(6, 72 mod 6) = HCF(6, 0)
HCF(6, 0) = 6 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(6, 120)

HCF(120, 6) = HCF(6, 120 mod 6) = HCF(6, 0)
HCF(6, 0) = 6 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 6, 72 and 120 is 6.

☛ Also Check:

HCF of 6, 72 and 120 Examples

Example 1: Verify the relation between the LCM and HCF of 6, 72 and 120.

Solution:

The relation between the LCM and HCF of 6, 72 and 120 is given as, HCF(6, 72, 120) = [(6 × 72 × 120) × LCM(6, 72, 120)]/[LCM(6, 72) × LCM (72, 120) × LCM(6, 120)]
⇒ Prime factorization of 6, 72 and 120:
- 6 = 2 × 3
- 72 = 2 × 2 × 2 × 3 × 3
- 120 = 2 × 2 × 2 × 3 × 5
∴ LCM of (6, 72), (72, 120), (6, 120), and (6, 72, 120) is 72, 360, 120, and 360 respectively.
Now, LHS = HCF(6, 72, 120) = 6.
And, RHS = [(6 × 72 × 120) × LCM(6, 72, 120)]/[LCM(6, 72) × LCM (72, 120) × LCM(6, 120)] = [(51840) × 360]/[72 × 360 × 120]
LHS = RHS = 6.
Hence verified.
Example 2: Find the highest number that divides 6, 72, and 120 completely.

Solution:

The highest number that divides 6, 72, and 120 exactly is their highest common factor.
- Factors of 6 = 1, 2, 3, 6
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
The HCF of 6, 72, and 120 is 6.
∴ The highest number that divides 6, 72, and 120 is 6.
Example 3: Calculate the HCF of 6, 72, and 120 using LCM of the given numbers.

Solution:

Prime factorization of 6, 72 and 120 is given as,
- 6 = 2 × 3
- 72 = 2 × 2 × 2 × 3 × 3
- 120 = 2 × 2 × 2 × 3 × 5
LCM(6, 72) = 72, LCM(72, 120) = 360, LCM(120, 6) = 120, LCM(6, 72, 120) = 360
⇒ HCF(6, 72, 120) = [(6 × 72 × 120) × LCM(6, 72, 120)]/[LCM(6, 72) × LCM (72, 120) × LCM(120, 6)]
⇒ HCF(6, 72, 120) = (51840 × 360)/(72 × 360 × 120)
⇒ HCF(6, 72, 120) = 6.
Therefore, the HCF of 6, 72 and 120 is 6.

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FAQs on HCF of 6, 72 and 120

What is the HCF of 6, 72 and 120?

The HCF of 6, 72 and 120 is 6. To calculate the highest common factor (HCF) of 6, 72 and 120, we need to factor each number (factors of 6 = 1, 2, 3, 6; factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; 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 6, 72 and 120, i.e., 6.

How to Find the HCF of 6, 72 and 120 by Prime Factorization?

To find the HCF of 6, 72 and 120, we will find the prime factorization of given numbers, i.e. 6 = 2 × 3; 72 = 2 × 2 × 2 × 3 × 3; 120 = 2 × 2 × 2 × 3 × 5.
⇒ Since 2, 3 are common terms in the prime factorization of 6, 72 and 120. Hence, HCF(6, 72, 120) = 2 × 3 = 6
☛ What is a Prime Number?

Which of the following is HCF of 6, 72 and 120? 6, 139, 163, 121, 165

HCF of 6, 72, 120 will be the number that divides 6, 72, and 120 without leaving any remainder. The only number that satisfies the given condition is 6.

What are the Methods to Find HCF of 6, 72 and 120?

There are three commonly used methods to find the HCF of 6, 72 and 120.

By Prime Factorization
By Euclidean Algorithm
By Long Division

What is the Relation Between LCM and HCF of 6, 72 and 120?

The following equation can be used to express the relation between LCM and HCF of 6, 72 and 120, i.e. HCF(6, 72, 120) = [(6 × 72 × 120) × LCM(6, 72, 120)]/[LCM(6, 72) × LCM (72, 120) × LCM(6, 120)].
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