1. What is GCF and LCM?

The GCF (Greatest Common Factor) is the largest number that divides two numbers without leaving a remainder. The LCM (Least Common Multiple) is the smallest number that is a multiple of both numbers.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

• \( a \) — First integer
• \( b \) — Second integer
• \( \gcd(a,b) \) — Greatest common divisor of a and b

Explanation: The GCF is calculated using the Euclidean algorithm, and the LCM is derived from the GCF using the relationship between these two values.

3. Importance of GCF and LCM

Details: GCF and LCM are fundamental concepts in number theory with applications in simplifying fractions, solving equations, and finding common denominators.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will compute both the GCF and LCM using efficient algorithms.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between GCF and GCD?
A: They are the same concept - GCD stands for Greatest Common Divisor, while GCF stands for Greatest Common Factor.

Q2: Can the calculator handle large numbers?
A: Yes, within reasonable limits of integer storage in PHP (typically up to 2^31-1).

Q3: What if I enter zero or negative numbers?
A: The calculator requires positive integers. Zero or negative inputs are not valid for GCF/LCM calculations.

Q4: How is the Euclidean algorithm efficient?
A: It reduces the problem size exponentially, making it very efficient even for large numbers.

Q5: Can I calculate GCF/LCM for more than two numbers?
A: This calculator handles two numbers, but the concepts can be extended to more numbers by iteratively applying the same methods.