Common Factor Calculator
This calculator finds every common factor of two, three or four whole numbers, works out the greatest common factor (GCF, also called the HCF), and calculates the least common multiple (LCM), all with the full working shown. Enter your numbers into the first two fields, plus a third and fourth if you need them, and the tool lists the factors of each number, highlights the ones they share, and returns the greatest common factor at the top of the results. You also get a count of how many common factors there are, the complete list of shared divisors, and the least common multiple, worked out from the relationship between GCF and LCM. Because the first two numbers are often a fraction, the calculator also shows that fraction simplified by dividing both the numerator and denominator by their GCF. Underneath the results, a step-by-step section walks through the Euclidean algorithm calculation used to reach the GCF, so you can see exactly how the answer was reached rather than just the final number. This makes the tool useful for simplifying fractions, finding a common denominator when adding or subtracting fractions, factoring algebraic expressions, or splitting a quantity into the largest possible equal groups. It works with positive whole numbers only, rounding any decimal entries to the nearest integer, and handles numbers up to 10,000,000 with exact results regardless of size.
1. Enter Your Numbers
2. Results
Factors of 48
Factors of 36
Step-by-Step Working
Factor Summary
Fraction Simplification
What Are Common Factors?
A factor of a number is any whole number that divides it evenly with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these divides into 12 without leaving a remainder. A common factor of two or more numbers is a factor that appears in the factor list of every one of those numbers. The greatest common factor (GCF) is simply the largest of these shared factors.
The GCF is also called the highest common factor (HCF) or greatest common divisor (GCD) depending on the country and context. In New Zealand, both GCF and HCF are used in mathematics education.
How This Calculator Works
This calculator uses the Euclidean algorithm to find the GCF of the numbers you enter. Once the GCF is known, every divisor of the GCF is also a common factor of the original numbers. The steps are:
- Find the GCF of the first two numbers using the Euclidean algorithm.
- If more numbers are entered, find the GCF of that result and the next number, repeating until all numbers are included.
- List all factors of the final GCF. These are all the common factors of the original numbers.
- Calculate the LCM using the relationship: LCM(a, b) = (a x b) / GCF(a, b).
The Euclidean Algorithm
The Euclidean algorithm is one of the oldest known algorithms, described by the Greek mathematician Euclid around 300 BC. It finds the GCF of two numbers by repeated division:
- Divide the larger number by the smaller number and find the remainder.
- Replace the larger number with the smaller number, and the smaller number with the remainder.
- Repeat until the remainder is zero. The last non-zero remainder is the GCF.
Example: GCF(48, 36). Step 1: 48 divided by 36 gives remainder 12. Step 2: 36 divided by 12 gives remainder 0. The GCF is 12.
Worked Example: Matching the Default Inputs
For the numbers 48 and 36:
| Step | Division | Remainder |
|---|---|---|
| 1 | 48 / 36 | 12 |
| 2 | 36 / 12 | 0 |
GCF = 12. The factors of 12 are: 1, 2, 3, 4, 6, 12. All six of these are common factors of 48 and 36. The LCM = (48 x 36) / 12 = 1,728 / 12 = 144. The fraction 48/36 simplifies to 4/3 (dividing both by the GCF of 12).
Why Common Factors Matter
Common factors are used throughout mathematics:
- Simplifying fractions: Divide both numerator and denominator by their GCF to get the simplest form. For example, 48/36 becomes 4/3.
- Adding and subtracting fractions: Find the LCM of the denominators to find the lowest common denominator.
- Factoring algebraic expressions: The GCF is factored out when simplifying polynomials.
- Solving problems involving groups: If you need to split objects into equal groups, the GCF tells you the largest possible group size.
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Method: GCF calculated using the Euclidean algorithm (division form). LCM derived from the identity LCM(a, b) = (a x b) / GCF(a, b), extended to multiple numbers by iteration. All factors of the GCF are enumerated by trial division from 1 to sqrt(GCF).
This calculator works with positive whole numbers only. Inputs are rounded to the nearest integer. For very large numbers (above 10,000,000), results remain exact as the Euclidean algorithm is not affected by number size.