1. What is GCD and LCM?

The GCD (Greatest Common Divisor) of two numbers is the largest number that divides both of them without leaving a remainder. The LCM (Least Common Multiple) is the smallest positive integer that is divisible by both numbers.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm to find the GCD, then calculates the LCM using the relationship:

Euclidean Algorithm Steps:

1. Divide the larger number by the smaller number
2. Find the remainder
3. Replace the larger number with the smaller number and the smaller number with the remainder
4. Repeat until the remainder is 0
5. The non-zero number at this point is the GCD

3. Importance of GCD and LCM

Applications: GCD is used to simplify fractions to their lowest terms. LCM is useful for finding common denominators in fractions, scheduling problems, and solving Diophantine equations.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will show the GCD and LCM along with the step-by-step work using the Euclidean algorithm.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between GCD and GCF?
A: GCD (Greatest Common Divisor) and GCF (Greatest Common Factor) are the same concept with different names.

Q2: Can this calculator handle more than two numbers?
A: This version calculates GCD and LCM for two numbers. For multiple numbers, you would need to calculate pairwise.

Q3: What's the time complexity of the Euclidean algorithm?
A: The Euclidean algorithm runs in O(log(min(a,b))) time, making it very efficient.

Q4: Can GCD be larger than the input numbers?
A: No, GCD cannot be larger than the smaller of the two input numbers.

Q5: What's the relationship between GCD and LCM?
A: For any two numbers a and b: GCD(a,b) × LCM(a,b) = a × b