Math & Engineering
Factor Calculator
Find all factors (divisors) of any positive integer number
Enter a positive integer to see its factors
Related to Factor Calculator
The factor calculator finds all numbers that divide evenly into a given positive integer. These numbers are called factors or divisors. For example, if you enter the number 12, the calculator will find all numbers that divide into 12 without leaving a remainder. The calculator uses an efficient algorithm to find all factors and organizes them in different ways to help you understand the number's properties.
Key Terms
- Factors: Numbers that divide evenly into another number
- Proper Factors: All factors of a number except the number itself
- Factor Pairs: Two numbers that multiply to give the original number
The calculator uses a systematic approach to find factors by checking all potential divisors up to the square root of the input number. This method is mathematically proven to find all factors efficiently. For each factor found, its corresponding pair is also identified, ensuring no factors are missed.
The calculator provides comprehensive information about the factors of your number, organized in several ways to help you understand its properties:
Results Explained
- All Factors: Complete list of numbers that divide evenly into your number
- Factor Pairs: Numbers grouped in pairs that multiply to give your number
- Number of Factors: Total count of factors
- Sum of Factors: The sum of all factors
- Proper Factors: All factors except the number itself
- Number of Proper Factors: Count of proper factors
- Sum of Proper Factors: Sum of all proper factors
These results are particularly useful in number theory, finding common factors between numbers, and understanding the properties of perfect numbers, abundant numbers, and deficient numbers based on the sum of proper factors.
1. What is a factor of a number?
A factor is a number that divides evenly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides into 12 without leaving a remainder.
2. What are proper factors?
Proper factors are all the factors of a number except the number itself. For example, the proper factors of 12 are 1, 2, 3, 4, and 6 (excluding 12). These are important in determining whether a number is perfect, abundant, or deficient.
3. Why are factor pairs important?
Factor pairs show which numbers multiply together to give the original number. They are useful in problem-solving, factoring algebraic expressions, and finding the dimensions of rectangles with a given area. For example, the factor pairs of 12 are (1,12), (2,6), and (3,4).
4. What is the relationship between factors and divisibility?
A number is divisible by its factors. If a is a factor of b, then b ÷ a will always result in a whole number. This relationship is fundamental in number theory and is used extensively in algebra and arithmetic calculations.
5. What is the scientific source for this calculator?
This calculator is based on fundamental number theory principles established in mathematics. The algorithm uses the well-documented divisibility test method from number theory, which states that all factors of a number n can be found by checking divisors up to √n. This method is proven in number theory textbooks and is referenced in mathematical literature, including Hardy and Wright's "An Introduction to the Theory of Numbers" and Kenneth H. Rosen's "Elementary Number Theory and Its Applications." The implementation follows standard mathematical algorithms for factor finding, ensuring accuracy and efficiency in calculations.