Introduction to the Highest Common Factor (HCF)

The highest common factor, also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more given integers without leaving a remainder. In this article, we will explore how to find the HCF of two specific numbers, 308 and 275, step-by-step.

What is the HCF of 308 and 275?

The HCF of 308 and 275 is 11. This means that 11 is the highest integer that can divide both 308 and 275 without leaving a remainder. Let's delve into the process of finding this HCF.

Step-by-Step Process of Finding the HCF of 308 and 275

1. Prime Factorization

The first step in finding the HCF is to perform the prime factorization of both numbers. Prime factorization involves breaking down a number into its prime factors. Here’s how to do it for 308 and 275:

308

Start with the smallest prime number, 2. 308 ÷ 2 154 154 ÷ 2 77 Move to the next smallest prime number, 7. 77 ÷ 7 11 11 is a prime number, so we stop here.

The prime factorization of 308 is 2×2×7×11.

275

Start with the smallest prime number, 5. 275 ÷ 5 55 55 ÷ 5 11 11 is a prime number, so we stop here.

The prime factorization of 275 is 5×5×11.

2. Identification of Common Factors

Now that we have the prime factorization of both numbers, we need to identify the common prime factors. From the factorizations, we can see:

Prime factorization of 308: 2×2×7×11 Prime factorization of 275: 5×5×11

The only common prime factor between 308 and 275 is 11. Since 11 is the highest number that appears in both factorizations, it is the HCF of 308 and 275.

Why 11 is the HCF

Given the step-by-step breakdown, we can see that 11 is the highest common factor in both numbers. This is because it is the largest prime number that divides both 308 and 275 without leaving a remainder.

Practical Applications of the HCF

The knowledge of the HCF is not just theoretical but has several practical applications. Some of these include:

Reducing fractions to their lowest terms Dividing items into smaller groups of equal size Simplifying complex mathematical problems

Understanding the HCF can help in simplifying these processes, making them more efficient and accurate.

Conclusion

In this article, we explored the highest common factor of 308 and 275, which is 11, through the process of prime factorization. By breaking down the numbers into their prime factors, we were able to identify the common factors and determine the HCF. The concept of HCF is essential in various fields, including mathematics, engineering, and data analysis.

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