1. What are GCF and LCM?

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

2. How Does the Calculator Work?

The calculator uses the mathematical formulas:

Where:

• \( a \) — First integer value
• \( b \) — Second integer value
• GCF — Greatest common factor calculated using Euclidean algorithm
• LCM — Least common multiple derived from GCF

Explanation: The Euclidean algorithm efficiently finds the GCF by repeatedly applying the modulo operation, then LCM is calculated using the relationship between GCF and LCM.

3. Importance of GCF and LCM

Details: GCF and LCM calculations are fundamental in number theory, useful for simplifying fractions, solving Diophantine equations, and finding common denominators in arithmetic operations.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will compute both GCF and LCM values simultaneously. Both values must be positive integers greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the Euclidean algorithm?
A: An efficient method for computing the greatest common divisor of two numbers, based on the principle that the GCF of two numbers also divides their difference.

Q2: Can GCF be larger than the input numbers?
A: No, the GCF cannot exceed the smaller of the two input numbers since it must divide both numbers.

Q3: What is the relationship between GCF and LCM?
A: For any two positive integers, GCF(a,b) × LCM(a,b) = a × b. This relationship allows efficient calculation of one from the other.

Q4: Can this calculator handle negative numbers?
A: No, this calculator only accepts positive integers since GCF and LCM are typically defined for natural numbers.

Q5: What about more than two numbers?
A: This calculator is designed for two numbers. For multiple numbers, you would need to compute GCF and LCM iteratively.