Greatest Common Factor Calculator
Solve the Greatest Common Factor (GCF) or Greatest Common Divisor (GCD) for a list of integers. Explore divisions step-by-step with Euclidean algorithms and prime factoring.
Enter between 2 and 10 positive whole numbers separated by commas. Max range: 100,000 per integer.
Greatest Common Factor (GCF)
12
Logistics Packaging Simulator
If these quantities represent different items to pack, you can package them into a maximum of 12 identical packets with no leftover scraps:
The Euclidean algorithm calculates GCF of multiple integers by recursively resolving the GCF of pairs: GCD(A, B, C) = GCD(GCD(A, B), C).
Understanding the Greatest Common Factor (GCF)
In arithmetic and number theory, the **Greatest Common Factor** (GCF), also commonly referred to as the **Greatest Common Divisor** (GCD), is the largest positive integer that divides each of the integers in a set without leaving a remainder.
Prime Factorization vs. Euclidean Algorithm
Our calculator implements the two primary mathematical strategies used to solve for GCF:
- Prime Factorization Method: Entails breaking each number down into its prime products, then multiplying the shared prime factors raised to their lowest powers. For example, $24 = 2^3 \times 3^1$ and $36 = 2^2 \times 3^2$. The shared primes are 2 and 3, and their lowest exponents are $2^2$ and $3^1$. Thus, $\text{GCF} = 2^2 \times 3^1 = 12$.
- Euclidean Division Algorithm: An ancient, highly efficient procedure that finds the GCD of two numbers by repeatedly dividing the larger number by the smaller number and calculating the remainder. This remainder replaces the larger number, and the division continues until the remainder reaches zero. The final non-zero remainder is the GCF.
Practical Applications
GCF calculations are extremely useful for real-world scenarios:
- Logistics Packaging: Distributing different quantities of inventory into the maximum number of identical packets without scraps.
- Flooring / Tiling: Finding the largest square tiles that can perfectly cover a rectangular room without having to make any cuts.