Common Factor Calculator — Free Online Common Divisor Finder

How to Use the Common Factor Calculator

1. Enter the first number: Type a positive integer into the first input field. The default is 36, which has many factors to demonstrate the tool's capabilities.
2. Enter the second number: Type a positive integer into the second input field. The default is 48. Together, 36 and 48 share several common factors.
3. Compare individual factors: Below the inputs, the calculator displays all factors of each number side by side. This visual comparison helps you understand which factors are shared and which are unique to each number.
4. Read the common factors: The results panel shows all common factors as visual badges. The Greatest Common Factor (GCF) is highlighted with a distinct color and label, making it immediately identifiable.
5. Review the summary: The bottom section provides a complete divisibility statement confirming that both numbers are divisible by all listed common factors.

Results update in real time as you change either input, making it easy to explore how common factors change with different number pairs. Try entering coprime numbers like 8 and 15 to see that their only common factor is 1.

Common Factor Formula

Variables Explained

• a, b: The two positive integers whose common factors you want to find.
• d: Any common divisor — a positive integer that divides both a and b exactly (zero remainder).
• GCF(a, b): The Greatest Common Factor — the largest d in the common factors set. All common factors are factors of the GCF.

Step-by-Step Example

Find all common factors of 60 and 84:

1. Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
2. Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
3. Common factors (in both lists): 1, 2, 3, 4, 6, 12
4. GCF = 12 (the largest common factor)

Verification: the factors of 12 are 12, which matches our common factors list exactly. This confirms the property that common factors of two numbers are always the factors of their GCF.

Practical Examples

Example 1: Emma's Craft Project

Emma has two ribbons: one 36 inches long and another 48 inches long. She wants to cut both into equal-length pieces with no waste. She needs to know all possible piece lengths.

• Common factors of 36 and 48: 1, 2, 3, 4, 6, 12
• Possible piece lengths: 1, 2, 3, 4, 6, or 12 inches
• At 12 inches (GCF): 36/12 = 3 pieces + 48/12 = 4 pieces = 7 total pieces
• At 6 inches: 36/6 = 6 pieces + 48/6 = 8 pieces = 14 total pieces

Emma has 6 different piece length options. The GCF of 12 inches gives the fewest, longest pieces, while smaller common factors give more, shorter pieces. The common factors represent every possible uniform length that works for both ribbons without waste.

Example 2: Carlos's Team Formation

Carlos is organizing a sports event with 42 boys and 56 girls. He wants to form equal-sized teams where each team has the same number of boys and the same number of girls. He needs to know all valid team sizes.

• Common factors of 42 and 56: 1, 2, 7, 14
• 14 teams: 3 boys + 4 girls per team
• 7 teams: 6 boys + 8 girls per team
• 2 teams: 21 boys + 28 girls per team

Carlos can form 2, 7, or 14 teams (or 1 combined team of everyone). The common factors tell him exactly which team counts allow perfectly equal distribution of both boys and girls. The GCF of 14 gives the maximum number of teams, which is ideal for organized competition.

Example 3: Sophie's Pattern Design

Sophie is designing a tiled floor pattern that repeats every 18 centimeters horizontally and every 24 centimeters vertically. She wants to find square tile sizes that align perfectly with both repeat distances.

• Common factors of 18 and 24: 1, 2, 3, 6
• Valid square tile sizes: 1, 2, 3, or 6 cm
• With 6 cm tiles: 18/6 = 3 tiles across, 24/6 = 4 tiles down per pattern unit

Sophie chooses 6 cm tiles (the GCF) for the simplest layout — 3 columns by 4 rows per pattern repeat. Using smaller common factors like 3 cm would also work but require more tiles. This type of problem appears in architecture, textile design, and computer graphics where repeating patterns must align with grid structures.

Common Factors Reference Table

Tips and Complete Guide

The Fundamental Property

The most important property of common factors is that they are exactly the factors of the GCF. This means you can find all common factors by first computing the GCF (using the efficient Euclidean algorithm), then listing all factors of the GCF. This two-step approach is far more efficient than listing all factors of both numbers and comparing them, especially for large numbers where the factor lists are extensive but the GCF may be relatively small.

Coprime Numbers

When two numbers have only 1 as their common factor, they are coprime. Surprisingly common in practice, about 61% of all random integer pairs are coprime (this relates to the fact that 6/pi^2 is approximately 0.608). Coprime numbers are essential in modular arithmetic — the Chinese Remainder Theorem works when the moduli are pairwise coprime. In practice, checking coprimality is as fast as computing the GCF.

Common Factors and Fraction Simplification

Any common factor can be used to partially simplify a fraction. For 24/36 with common factors 12: dividing by 2 gives 12/18, by 3 gives 8/12, by 4 gives 6/9, by 6 gives 4/6, by 12 gives 2/3. Only dividing by the GCF (12) gives the fully simplified form in one step. However, sometimes dividing by smaller common factors first makes mental arithmetic easier — dividing 360/480 by 10 first (to get 36/48) is much easier than mentally dividing by 120 (the GCF).

Venn Diagram Visualization

A Venn diagram is an excellent way to visualize common factors. Place all factors of number A in the left circle and all factors of number B in the right circle. The overlap (intersection) contains the common factors. This visual approach makes the concept intuitive, especially when teaching the relationship between individual factors, common factors, and the GCF. The GCF is simply the largest number in the intersection.

Common Mistakes to Avoid

• Confusing common factors with common multiples: Common factors are shared divisors (smaller than both numbers), while common multiples are shared products (larger than both numbers). They are related through the GCF-LCM formula but are conceptually different.
• Forgetting to include 1: The number 1 is always a common factor of any two positive integers. It should always be included in the common factors list.
• Missing factors when listing manually: When finding factors of each number individually, it is easy to skip some, especially for numbers with many factors. Use systematic trial division up to the square root to avoid missing any.
• Assuming larger numbers have more common factors: The number of common factors depends on the GCF, not the magnitude of the numbers. GCF(100, 1000) = 100 has 9 common factors, but GCF(97, 101) = 1 has only 1 common factor, despite both pairs involving large numbers.
• Thinking coprime means prime: Coprime is a property of a pair of numbers, meaning they share no common factors beyond 1. Coprime numbers need not themselves be prime — 8 and 15 are both composite but coprime as a pair.

Frequently Asked Questions