1. What is GCD?

The GCD (Greatest Common Divisor) of three numbers is the largest positive integer that divides each of the numbers without leaving a remainder. For example, gcd(3,1,3) = 1.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm:

Where:

• \( a, b, c \) — Positive integers
• gcd(a,b) — Greatest common divisor of a and b

Explanation: The Euclidean algorithm works by repeatedly dividing the larger number by the smaller number and using the remainder until the remainder is zero.

3. Importance of GCD Calculation

Details: GCD is fundamental in number theory and has applications in simplifying fractions, cryptography, and algorithm design.

4. Using the Calculator

Tips: Enter three positive integers. The calculator will find their greatest common divisor.

5. Frequently Asked Questions (FAQ)

Q1: What is the GCD of prime numbers?
A: The GCD of two or more prime numbers is 1, unless they are the same prime number.

Q2: Can GCD be calculated for more than three numbers?
A: Yes, the same method can be extended to any number of integers by iteratively calculating GCDs.

Q3: What is the relationship between GCD and LCM?
A: For two numbers, gcd(a,b) × lcm(a,b) = a × b. This doesn't extend directly to three or more numbers.

Q4: What's the GCD of zero and a number?
A: The GCD of 0 and any number a is |a|. However, our calculator requires positive integers.

Q5: How efficient is the Euclidean algorithm?
A: It's very efficient, with a time complexity of O(log min(a,b)) for two numbers.