Common Factor Calculator
Find all common factors (divisors) of two or more positive integers.
Understanding Common Factors
Common factors are positive integers that divide two or more numbers without a remainder. The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest such number.
What are Factors?
Factors (or divisors) of a number are integers that divide the number evenly without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
How to Find Common Factors:
- List all factors of each number individually
- Identify the factors that appear in all lists
- These shared factors are the common factors
- The largest number in this list is the Greatest Common Factor (GCF) or Greatest Common Divisor (GCD)
Examples of Common Factors:
Example 1: Find common factors of 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
- GCD = 6
Example 2: Find common factors of 24, 36, and 48
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Common factors: 1, 2, 3, 4, 6, 12
- GCD = 12
Applications of Common Factors:
In Mathematics:
- Simplifying fractions to lowest terms
- Solving Diophantine equations
- Computing the Least Common Multiple (LCM)
- Calculating proportions and ratios
- Finding common denominators
Practical Applications:
- Dividing items equally into groups
- Determining common intervals or cycles
- Simplifying measurements
- Calculating gear ratios
- Computer algorithm optimizations
Advanced Methods for Finding GCD
- Euclidean Algorithm: An efficient method that uses successive division to find the GCD without listing all factors.
- Prime Factorization: Finding the GCD by identifying common prime factors shared by all numbers.
- Stein's Algorithm: A binary GCD algorithm that replaces division with simpler operations like shifting and subtraction.
- Extended Euclidean Algorithm: An extension that computes the coefficients of Bézout's identity in addition to the GCD.
How to Use the Calculator:
- Enter two or more positive integers separated by commas in the input field
- Click "Find Common Factors" to calculate the result
- View all common factors and the Greatest Common Divisor (GCD)
- See additional statistics about the common factors