How do you find the greatest common factor (GCF)?

The Greatest Common Factor (GCF) is the largest number in the list of common factors. For 12 and 18, the GCF is 6. A highly efficient method for finding the GCF is the Euclidean algorithm, which uses repeated division and remainders.

How do prime factorizations help find common factors?

By finding the prime factorization of each number, you can identify the shared prime factors. The GCF is the product of these shared prime factors, raised to the lowest power they appear in any of the factorizations. For example, 12 = 2² × 3 and 18 = 2 × 3². The shared primes are 2 and 3. The lowest power of 2 is 2¹ and the lowest power of 3 is 3¹. So, the GCF is 2¹ × 3¹ = 6.

Can this tool list all common divisors for multiple numbers?

Yes. You can enter two or more numbers, separated by commas, spaces, or newlines. The calculator will compute the GCF of the entire set and then list all the divisors of that GCF, which are the common factors for the whole set of numbers.

Is batch factorization supported (CSV)?

Yes. You can paste multiple rows of numbers (e.g., from a CSV or spreadsheet). The calculator will treat each row as a separate set of numbers and compute the common factors and GCF for each set individually, displaying the results in a summary table.

Are my results saved or sent anywhere?

No. All calculations are performed entirely on your device, within your web browser. No data is ever sent to any server, ensuring your privacy.

Common Factor Calculator — Find Common Divisors, GCF, Shared Prime Factors & Step-by-Step Work

Common Factor Calculator

All calculations are done on your device.

What Are Common Factors?

Common factors, also known as common divisors, are numbers that can divide two or more different integers without leaving a remainder. Every set of integers has at least one common factor: the number 1. The goal is often to find all such factors, and especially the largest one.

For example, let's consider the numbers 12 and 18.

The factors of 12 are: {1, 2, 3, 4, 6, 12}.
The factors of 18 are: {1, 2, 3, 6, 9, 18}.

The numbers that appear in both lists are the common factors. In this case, the common factors of 12 and 18 are {1, 2, 3, 6}. The largest number in this set, 6, is called the Greatest Common Factor (GCF) or Greatest Common Divisor (GCD).

How to Find Common Factors — Methods Explained

There are two primary methods for finding the GCF, which is the key to finding all other common factors.

Prime Factorization Intersection: This method is intuitive and visual. You first find the prime factorization of each number.
- 12 = 2² × 3¹
- 18 = 2¹ × 3²
Next, you identify the prime factors that are common to all the numbers (in this case, 2 and 3). For each common prime, you take the one with the lowest exponent. Here, the lowest power of 2 is 2¹, and the lowest power of 3 is 3¹. Multiplying these together gives the GCF: 2¹ × 3¹ = 6. All common factors are simply the divisors of this GCF.
The Euclidean Algorithm: This is a highly efficient, ancient algorithm that finds the GCF of two numbers without needing to find their prime factors first. It works by repeated division:
- Divide 18 by 12: 18 = 1 × 12 + 6 (remainder is 6)
- Now, divide the previous divisor (12) by the remainder (6): 12 = 2 × 6 + 0 (remainder is 0)
The last non-zero remainder is the GCF. In this case, it's 6. This method is exceptionally fast, especially for very large numbers.

When to Use GCF and Common Factors

Finding common factors and the GCF is a fundamental skill in mathematics with many practical applications:

Simplifying Fractions: To reduce a fraction to its simplest form, you divide both the numerator and the denominator by their GCF. For example, to simplify 12/18, you divide both by GCF(12, 18) = 6 to get 2/3.
Solving Ratios: Similar to fractions, simplifying ratios often involves dividing each part of the ratio by the GCF of all parts.
Calculating the LCM: The Least Common Multiple (LCM) is easily found using the GCF. The formula is LCM(a, b) = (|a × b|) / GCF(a, b).
Real-World Problems: Problems involving arranging items into equal groups, tiling floors with same-sized squares, or finding when cyclical events will align often require finding the GCF or LCM.

Frequently Asked Questions

What are the common factors if one number is zero?: The set of common factors of 0 and any non-zero integer 'n' is simply the set of all factors of 'n'. This is because every integer is a factor of 0. Therefore, the GCF(0, n) is |n|.
How are negative numbers handled?: Factors are typically considered positive integers. This calculator finds the common factors of the absolute (positive) values of the input numbers.
What if the numbers are coprime?: Two numbers are "coprime" or "relatively prime" if their only common factor is 1. In this case, their GCF is 1. For example, 8 and 15 are coprime.
How do I find common factors for more than two numbers?: You first find the GCF of the entire set of numbers. You can do this by repeatedly applying the Euclidean algorithm: GCF(a, b, c) = GCF(GCF(a, b), c). Once you have the final GCF, the set of all its divisors will be the common factors for the entire original set of numbers.

Disclaimer

This Common Factor Calculator is designed as an educational tool to help visualize and understand the relationships between integers. It uses robust algorithms to handle large numbers efficiently. For mission-critical applications or tasks involving cryptography, please consult specialized, professionally audited software libraries.