GCD Calculator
Find the Greatest Common Divisor of two integers instantly using the Euclidean Algorithm.
GCD:
What is the Greatest Common Divisor (GCD)?
The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF) or Greatest Common Factor (GCF), is the largest positive integer that divides each of two or more integers without leaving a remainder. For example, the GCD of 8 and 12 is 4.
How to Calculate GCD
While there are several methods to find the GCD, such as listing all factors or using prime factorization, the most efficient method for large numbers is the Euclidean Algorithm. This algorithm uses the principle that the GCD of two numbers also divides their difference.
Example Calculation (GCD of 48 and 18):
1. Divide 48 by 18: 48 = (2 × 18) + 12
2. Divide 18 by 12: 18 = (1 × 12) + 6
3. Divide 12 by 6: 12 = (2 × 6) + 0
4. Since the remainder is 0, the GCD is 6.
1. Divide 48 by 18: 48 = (2 × 18) + 12
2. Divide 18 by 12: 18 = (1 × 12) + 6
3. Divide 12 by 6: 12 = (2 × 6) + 0
4. Since the remainder is 0, the GCD is 6.
Why is GCD Important?
The GCD is a fundamental concept in number theory and has several practical applications:
- Simplifying Fractions: To reduce a fraction to its lowest terms, divide both the numerator and the denominator by their GCD.
- Finding the Least Common Multiple (LCM): The relationship between GCD and LCM is given by the formula: (a × b) / GCD(a, b) = LCM(a, b).
- Cryptography: GCD calculations are core to modern encryption algorithms like RSA, which keep digital communications secure.
- Geometry: Determining the largest square tile size that can perfectly cover a rectangular floor.
GCD Properties
Understanding these properties can help solve complex math problems quickly:
- The GCD of any number n and 0 is n.
- The GCD of two prime numbers is always 1.
- If the GCD of two numbers is 1, they are called coprime or relatively prime.