Hcf Of 6615 And 9702

Finding the Highest Common Factor (HCF) of 6615 and 9702: A practical guide

Finding the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two numbers is a fundamental concept in number theory. This article will guide you through various methods to determine the HCF of 6615 and 9702, explaining the underlying principles and providing a deeper understanding of the process. And we'll explore different approaches, from prime factorization to the Euclidean algorithm, ensuring you grasp the concepts thoroughly. Understanding HCF is crucial in various mathematical applications, including simplifying fractions, solving algebraic problems, and understanding modular arithmetic.

Introduction to Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. In simpler terms, it's the biggest number that's a factor of both numbers. As an example, the HCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly. Finding the HCF is a valuable skill in mathematics and has numerous applications in various fields.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Think about it: a prime number is a whole number greater than 1 that has only two divisors: 1 and itself. The prime factors are the prime numbers that, when multiplied together, give the original number. Let's apply this method to find the HCF of 6615 and 9702.

1. Prime Factorization of 6615:

We start by finding the prime factors of 6615. We can use a factor tree or repeated division by prime numbers.

• 6615 is divisible by 3: 6615 = 3 × 2205
• 2205 is divisible by 3: 2205 = 3 × 735
• 735 is divisible by 3: 735 = 3 × 245
• 245 is divisible by 5: 245 = 5 × 49
• 49 is divisible by 7: 49 = 7 × 7

That's why, the prime factorization of 6615 is 3³ × 5 × 7².

2. Prime Factorization of 9702:

Now let's find the prime factorization of 9702.

• 9702 is divisible by 2: 9702 = 2 × 4851
• 4851 is divisible by 3: 4851 = 3 × 1617
• 1617 is divisible by 3: 1617 = 3 × 539
• 539 is divisible by 7: 539 = 7 × 77
• 77 is divisible by 7: 77 = 7 × 11

Which means, the prime factorization of 9702 is 2 × 3² × 7² × 11.

3. Finding the HCF:

To find the HCF, we identify the common prime factors in both factorizations and multiply them together, taking the lowest power of each common factor.

Both numbers have 3, 7 as common factors. The lowest power of 3 is 3¹ (or 3) and the lowest power of 7 is 7².

Which means, the HCF of 6615 and 9702 is 3 × 7² = 3 × 49 = 147.

Method 2: Euclidean Algorithm

The Euclidean algorithm is a more efficient method for finding the HCF of two numbers, especially for larger numbers. It's based on the principle that the HCF of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, and that number is the HCF.

Steps:

1. Divide the larger number by the smaller number and find the remainder.
2. Replace the larger number with the smaller number and the smaller number with the remainder.
3. Repeat steps 1 and 2 until the remainder is 0.
4. The last non-zero remainder is the HCF.

Let's apply the Euclidean algorithm to find the HCF of 6615 and 9702:

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1. 9702 ÷ 6615 = 1 with a remainder of 3087.
2. 6615 ÷ 3087 = 2 with a remainder of 441.
3. 3087 ÷ 441 = 7 with a remainder of 0.

The last non-zero remainder is 441. Because of this, the HCF of 6615 and 9702 using the Euclidean Algorithm is 441.

Note: There seems to be a discrepancy between the results of the prime factorization method and the Euclidean algorithm. Let's re-examine the calculations. There was an error in the initial prime factorization. Let's correct it.

Recalculating Prime Factorization

Let's carefully re-do the prime factorization of 6615 and 9702.

Prime Factorization of 6615:

6615 = 3 x 2205 = 3 x 3 x 735 = 3 x 3 x 3 x 245 = 3 x 3 x 3 x 5 x 49 = 3³ x 5 x 7²

Prime Factorization of 9702:

9702 = 2 x 4851 = 2 x 3 x 1617 = 2 x 3 x 3 x 539 = 2 x 3² x 7 x 77 = 2 x 3² x 7² x 11

Finding the HCF (Corrected):

Common factors are 3 and 7. The lowest power of 3 is 3¹ = 3 and the lowest power of 7 is 7².

HCF = 3¹ x 7² = 3 x 49 = 147.

This still differs from the Euclidean algorithm result. Let's re-examine the Euclidean Algorithm.

Recalculating Euclidean Algorithm

1. 9702 ÷ 6615 = 1 remainder 3087
2. 6615 ÷ 3087 = 2 remainder 441
3. 3087 ÷ 441 = 7 remainder 0

The last non-zero remainder is 441. There's a significant discrepancy. Let's investigate further. The prime factorization method is more prone to errors if a prime factor is missed. The Euclidean algorithm is generally more reliable for larger numbers. Let's check the prime factorization of 441.

441 = 3² x 7²

This means 441 is a factor of both 6615 and 9702. Let's check this:

6615 / 441 = 15 9702 / 441 = 22

The Euclidean Algorithm result of 441 is correct. The error was in the initial manual prime factorization. It is crucial to be meticulous when performing prime factorization manually.

Conclusion

The correct Highest Common Factor (HCF) of 6615 and 9702 is 441. Also, remember to always double-check your work to ensure accuracy. The Euclidean algorithm proved to be a more reliable method in this case, highlighting the importance of using multiple methods and carefully checking calculations when working with large numbers. While prime factorization provides a deeper understanding of the number's structure, the Euclidean algorithm offers a more efficient and less error-prone approach for determining the HCF. Using a combination of methods and verifying results provides confidence in your mathematical solutions.