Highest Common Factor Of 75 And 105

Finding the Highest Common Factor (HCF) of 75 and 105

This article will guide you through different methods to find the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of 75 and 105. Understanding HCF is crucial in various mathematical applications, from simplifying fractions to solving algebraic problems. We'll explore prime factorization and the Euclidean algorithm, offering a comprehensive understanding of this fundamental concept.

What is the Highest Common Factor (HCF)?

The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In simpler terms, it's the biggest number that goes into both numbers evenly. For instance, the HCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 without leaving a remainder. Finding the HCF is a valuable skill in simplifying fractions and solving various mathematical problems.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves. Let's find the prime factorization of 75 and 105:

• 75: 3 x 5 x 5 = 3 x 5²
• 105: 3 x 5 x 7

Once we have the prime factorization of both numbers, we identify the common prime factors and their lowest powers. Both 75 and 105 share 3 and 5 as prime factors. The lowest power of 3 is 3¹ (or simply 3) and the lowest power of 5 is 5¹.

Therefore, the HCF of 75 and 105 is 3 x 5 = 15.

Method 2: The Euclidean Algorithm

The Euclidean algorithm is an efficient method, especially for larger numbers. It uses successive division until the remainder is 0. The last non-zero remainder is the HCF.

1. Divide the larger number (105) by the smaller number (75): 105 ÷ 75 = 1 with a remainder of 30

2. Replace the larger number with the smaller number (75) and the smaller number with the remainder (30): 75 ÷ 30 = 2 with a remainder of 15

3. Repeat the process: 30 ÷ 15 = 2 with a remainder of 0

Since the remainder is 0, the last non-zero remainder (15) is the HCF. Therefore, the HCF of 75 and 105 is 15.

Conclusion:

Both methods, prime factorization and the Euclidean algorithm, effectively determine the HCF. The prime factorization method provides a clear visual representation of the factors, while the Euclidean algorithm is more efficient for larger numbers. Understanding these methods empowers you to easily find the HCF of any two numbers, a fundamental skill in various mathematical contexts. Remember, the HCF represents the greatest common divisor, a concept essential for simplifying fractions and solving more complex mathematical problems involving divisibility and common factors.