1. What is the Greatest Common Factor?

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. It is a fundamental concept in number theory and mathematics.

2. How Does the Euclidean Algorithm Work?

The calculator uses the Euclidean Algorithm:

Where:

• \( a \) — First integer number
• \( b \) — Second integer number
• \( a \mod b \) — Remainder when a is divided by b

Explanation: The algorithm repeatedly replaces the larger number with the remainder of dividing the larger by the smaller, until the remainder becomes zero. The last non-zero remainder is the GCF.

3. Importance of GCF Calculation

Details: GCF is essential for simplifying fractions, solving Diophantine equations, cryptography, and various mathematical applications. It helps in reducing fractions to their simplest form and finding common denominators.

4. Using the Calculator

Tips: Enter two positive integers in the input fields. The calculator will compute their Greatest Common Factor using the efficient Euclidean algorithm method.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between GCF and LCM?
A: GCF (Greatest Common Factor) finds the largest number that divides both numbers, while LCM (Least Common Multiple) finds the smallest number that is a multiple of both numbers.

Q2: Can the Euclidean algorithm handle negative numbers?
A: The algorithm works with absolute values, so negative numbers are converted to positive since GCF is always positive.

Q3: What is the GCF of prime numbers?
A: If two numbers are prime and different, their GCF is 1. If they are the same prime number, their GCF is that prime number.

Q4: How does the algorithm work with zero?
A: GCF(a, 0) = |a|, and GCF(0, 0) is undefined. Our calculator requires positive integers.

Q5: What is the time complexity of Euclidean algorithm?
A: The Euclidean algorithm has O(log(min(a, b))) time complexity, making it very efficient even for large numbers.