Math & Engineering
Greatest Common Factor (GCF) Calculator
Calculate the greatest common factor (GCF) of two or more numbers. Also known as the greatest common divisor (GCD).
Enter at least two numbers to calculate their GCF
Related to Greatest Common Factor (GCF) Calculator
The Greatest Common Factor (GCF) calculator finds the largest positive number that divides evenly into two or more numbers. The calculator uses the efficient Euclidean algorithm to compute the GCF, also known as the Greatest Common Divisor (GCD).
Euclidean Algorithm
The calculator implements the Euclidean algorithm, which repeatedly divides the larger number by the smaller one and takes the remainder until reaching zero. The last non-zero remainder is the GCF. For multiple numbers, the algorithm is applied pairwise to find the overall GCF.
Example Calculation
To find the GCF of 48 and 18:
48 = 2 × 18 + 12
18 = 1 × 12 + 6
12 = 2 × 6 + 0
Therefore, GCF(48, 18) = 6
The GCF result represents the largest positive integer that divides evenly into all input numbers. This number has important applications in mathematics, particularly in fraction simplification and solving linear Diophantine equations.
Key Properties
• The GCF is always positive, even for negative input numbers
• If the GCF is 1, the numbers are called coprime or relatively prime
• The GCF of a number and zero is the absolute value of the non-zero number
• The GCF of two consecutive integers is always 1
1. What is the difference between GCF and GCD?
GCF (Greatest Common Factor) and GCD (Greatest Common Divisor) are two names for exactly the same mathematical concept. The term GCF is more commonly used in elementary mathematics, while GCD is more prevalent in higher mathematics and computer science.
2. Can the GCF calculator handle negative numbers?
Yes, the calculator can handle negative numbers. The GCF is always positive, as it is defined as the largest positive number that divides evenly into the given numbers. For example, GCF(-12, 18) = GCF(12, 18) = 6.
3. What happens if I enter zero as one of the numbers?
If one of the numbers is zero, the GCF will be the absolute value of the other number. This is because any number divides evenly into zero, so the largest such number is the absolute value of the non-zero number.
4. Why is finding the GCF important?
Finding the GCF is crucial for simplifying fractions, solving linear Diophantine equations, and finding common denominators. In practical applications, it helps in dividing quantities into equal groups, determining repeating patterns, and solving various mathematical problems.
5. What is the scientific source for this calculator?
This calculator implements the Euclidean algorithm, first described in Euclid's Elements (circa 300 BCE), Book VII, Propositions 1 and 2. The algorithm's mathematical proof and efficiency have been extensively studied in number theory. Modern implementations follow the standard algorithm as described in Donald Knuth's "The Art of Computer Programming, Volume 2: Seminumerical Algorithms" (1969), which provides a comprehensive analysis of the algorithm's computational complexity and mathematical properties.