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HCF of 106, 159 and 265

HCF of 106, 159 and 265 is the largest possible number that divides 106, 159 and 265 exactly without any remainder. The factors of 106, 159 and 265 are (1, 2, 53, 106), (1, 3, 53, 159) and (1, 5, 53, 265) respectively. There are 3 commonly used methods to find the HCF of 106, 159 and 265 - prime factorization, long division, and Euclidean algorithm.

1.	HCF of 106, 159 and 265
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 106, 159 and 265?

Answer: HCF of 106, 159 and 265 is 53.

Explanation:

The HCF of three non-zero integers, x(106), y(159) and z(265), is the highest positive integer m(53) that divides x(106), y(159) and z(265) without any remainder.

Methods to Find HCF of 106, 159 and 265

Let's look at the different methods for finding the HCF of 106, 159 and 265.

Long Division Method
Prime Factorization Method
Listing Common Factors

HCF of 106, 159 and 265 by Long Division

HCF of 106, 159 and 265 can be represented as HCF of (HCF of 106, 159) and 265. HCF(106, 159, 265) can be thus calculated by first finding HCF(106, 159) using long division and thereafter using this result with 265 to perform long division again.

Step 1: Divide 159 (larger number) by 106 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (106) by the remainder (53). Repeat this process until the remainder = 0.
⇒ HCF(106, 159) = 53.
Step 3: Now to find the HCF of 53 and 265, we will perform a long division on 265 and 53.
Step 4: For remainder = 0, divisor = 53 ⇒ HCF(53, 265) = 53

Thus, HCF(106, 159, 265) = HCF(HCF(106, 159), 265) = 53.

HCF of 106, 159 and 265 by Prime Factorization

Prime factorization of 106, 159 and 265 is (2 × 53), (3 × 53) and (5 × 53) respectively. As visible, 106, 159 and 265 have only one common prime factor i.e. 53. Hence, the HCF of 106, 159 and 265 is 53.

HCF of 106, 159 and 265 by Listing Common Factors

Factors of 106: 1, 2, 53, 106
Factors of 159: 1, 3, 53, 159
Factors of 265: 1, 5, 53, 265

There are 2 common factors of 106, 159 and 265, that are 1 and 53. Therefore, the highest common factor of 106, 159 and 265 is 53.

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HCF of 106, 159 and 265 Examples

Example 1: Verify the relation between the LCM and HCF of 106, 159 and 265.

Solution:

The relation between the LCM and HCF of 106, 159 and 265 is given as, HCF(106, 159, 265) = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(106, 265)]
⇒ Prime factorization of 106, 159 and 265:
- 106 = 2 × 53
- 159 = 3 × 53
- 265 = 5 × 53
∴ LCM of (106, 159), (159, 265), (106, 265), and (106, 159, 265) is 318, 795, 530, and 1590 respectively.
Now, LHS = HCF(106, 159, 265) = 53.
And, RHS = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(106, 265)] = [(4466310) × 1590]/[318 × 795 × 530]
LHS = RHS = 53.
Hence verified.
Example 2: Find the highest number that divides 106, 159, and 265 completely.

Solution:

The highest number that divides 106, 159, and 265 exactly is their highest common factor.
- Factors of 106 = 1, 2, 53, 106
- Factors of 159 = 1, 3, 53, 159
- Factors of 265 = 1, 5, 53, 265
The HCF of 106, 159, and 265 is 53.
∴ The highest number that divides 106, 159, and 265 is 53.
Example 3: Calculate the HCF of 106, 159, and 265 using LCM of the given numbers.

Solution:

Prime factorization of 106, 159 and 265 is given as,
- 106 = 2 × 53
- 159 = 3 × 53
- 265 = 5 × 53
LCM(106, 159) = 318, LCM(159, 265) = 795, LCM(265, 106) = 530, LCM(106, 159, 265) = 1590
⇒ HCF(106, 159, 265) = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(265, 106)]
⇒ HCF(106, 159, 265) = (4466310 × 1590)/(318 × 795 × 530)
⇒ HCF(106, 159, 265) = 53.
Therefore, the HCF of 106, 159 and 265 is 53.

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FAQs on HCF of 106, 159 and 265

What is the HCF of 106, 159 and 265?

The HCF of 106, 159 and 265 is 53. To calculate the HCF (Highest Common Factor) of 106, 159 and 265, we need to factor each number (factors of 106 = 1, 2, 53, 106; factors of 159 = 1, 3, 53, 159; factors of 265 = 1, 5, 53, 265) and choose the highest factor that exactly divides 106, 159 and 265, i.e., 53.

Which of the following is HCF of 106, 159 and 265? 53, 273, 304, 270, 275

HCF of 106, 159, 265 will be the number that divides 106, 159, and 265 without leaving any remainder. The only number that satisfies the given condition is 53.

How to Find the HCF of 106, 159 and 265 by Prime Factorization?

To find the HCF of 106, 159 and 265, we will find the prime factorization of given numbers, i.e. 106 = 2 × 53; 159 = 3 × 53; 265 = 5 × 53.
⇒ Since 53 is the only common prime factor of 106, 159 and 265. Hence, HCF(106, 159, 265) = 53.
☛ What are Prime Numbers?

What are the Methods to Find HCF of 106, 159 and 265?

There are three commonly used methods to find the HCF of 106, 159 and 265.

By Long Division
By Prime Factorization
By Euclidean Algorithm

What is the Relation Between LCM and HCF of 106, 159 and 265?

The following equation can be used to express the relation between Least Common Multiple and HCF of 106, 159 and 265, i.e. HCF(106, 159, 265) = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(106, 265)].
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