How to Calculate Greatest Common Divisor

Greatest Common Divisor (GCD) Calculator

function calculateGCD() { var num1Input = document.getElementById("firstNumber").value; var num2Input = document.getElementById("secondNumber").value; var resultDiv = document.getElementById("gcdResult"); var num1 = parseInt(num1Input); var num2 = parseInt(num2Input); if (isNaN(num1) || isNaN(num2) || num1 <= 0 || num2 <= 0) { resultDiv.innerHTML = "Please enter two positive whole numbers."; return; } // Euclidean algorithm to find GCD function findGCD(a, b) { while (b !== 0) { var temp = b; b = a % b; a = temp; } return a; } var gcdValue = findGCD(num1, num2); resultDiv.innerHTML = "The Greatest Common Divisor (GCD) of " + num1 + " and " + num2 + " is: " + gcdValue + ""; } // Calculate on page load with default values window.onload = calculateGCD;

Understanding the Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF), of two or more non-zero integers, is the largest positive integer that divides each of the integers without leaving a remainder. It's a fundamental concept in number theory with applications in various fields, including simplifying fractions, cryptography, and computer science.

Why is GCD Important?

• Simplifying Fractions: To reduce a fraction to its simplest form, you divide both the numerator and the denominator by their GCD. For example, the GCD of 12 and 18 is 6, so 12/18 simplifies to (12÷6)/(18÷6) = 2/3.
• Modular Arithmetic: GCD plays a role in solving linear Diophantine equations and understanding modular inverses.
• Cryptography: Algorithms like RSA rely on properties related to GCD and prime numbers.
• Computer Science: Used in algorithms for tasks like array manipulation and resource allocation.

How is GCD Calculated? The Euclidean Algorithm

While you can find the GCD by listing all divisors of each number and finding the largest common one, this method becomes cumbersome for larger numbers. The most efficient and widely used method is the Euclidean Algorithm.

The Euclidean Algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until one of the numbers becomes zero, and the other number is the GCD.

More formally, for two non-negative integers a and b (where a > b), the GCD of a and b is the same as the GCD of b and the remainder when a is divided by b (a mod b). This continues until the remainder is 0, at which point the non-zero number is the GCD.

Example Calculation: GCD of 48 and 18

1. Divide 48 by 18: 48 = 2 × 18 + 12 (Remainder is 12)
2. Now, replace the larger number (18) with the smaller number (12) and the smaller number (12) with the remainder (12): GCD(18, 12)
3. Divide 18 by 12: 18 = 1 × 12 + 6 (Remainder is 6)
4. Now, replace the larger number (12) with the smaller number (6) and the smaller number (6) with the remainder (6): GCD(12, 6)
5. Divide 12 by 6: 12 = 2 × 6 + 0 (Remainder is 0)

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 6.

Use the calculator above to quickly find the GCD for any two positive integers!