Using the Factor Calculator
The factor calculator is a versatile tool designed for students, teachers, and math enthusiasts to quickly find all divisors of a given integer. Whether you are simplifying fractions, finding the Greatest Common Factor (GCF), or exploring number theory, this calculator provides instant results including prime factorization and primality checks.
To use this tool, simply enter any positive whole number. You can choose from three primary calculation modes:
- Find All Factors
- This option lists every integer that divides into your chosen number without leaving a remainder. It also groups them into factor pairs.
- Prime Factorization
- This determines the set of prime numbers which, when multiplied together, equal the original number. It is displayed in both expanded and exponential formats.
- Check Primality
- Quickly determine if a number is prime (only divisible by 1 and itself) or composite.
How to Find Factors of a Number
A factor is a number that divides another number completely, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The mathematical formula for factoring is simple:
If a × b = n, then both a and b are factors of n.
To find factors manually, follow these steps:
- Start by dividing the number by 1. Since every number is divisible by 1, 1 and the number itself are always the first factor pair.
- Try dividing by 2. If the number is even, 2 is a factor.
- Continue testing subsequent integers (3, 4, 5…) to see if they divide the number evenly.
- Stop testing once you reach the square root of the number. Any factor larger than the square root will have already been paired with a factor smaller than the square root.
Prime Factorization Explained
Prime factorization is the process of breaking down a composite number into its basic building blocks—prime numbers. According to the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization.
Example: Find the prime factors of 60.
- 60 is even, so divide by 2: 60 = 2 × 30
- 30 is even, so divide by 2: 30 = 2 × 15
- 15 is divisible by 3: 15 = 3 × 5
- 5 is a prime number.
- The prime factorization is: 2 × 2 × 3 × 5, or 22 × 3 × 5.
Factorization Example Table
Below are common numbers and their factors as calculated by our factor calculator:
| Number | All Factors | Prime Factorization |
|---|---|---|
| 16 | 1, 2, 4, 8, 16 | 24 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 23 × 3 |
| 37 | 1, 37 | Prime |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 22 × 52 |
Divisibility Rules for Factoring
When using the factor calculator manually, you can use these shortcuts to find divisors quickly:
- By 2: The last digit is even (0, 2, 4, 6, 8).
- By 3: The sum of the digits is divisible by 3.
- By 4: The last two digits are divisible by 4.
- By 5: The last digit is 0 or 5.
- By 6: The number is divisible by both 2 and 3.
- By 9: The sum of the digits is divisible by 9.
- By 10: The number ends in 0.
Common Questions
What is the difference between a factor and a multiple?
A factor is a number that divides evenly into another (e.g., 3 is a factor of 12). A multiple is the result of multiplying a number by an integer (e.g., 12 is a multiple of 3). Factors are always equal to or smaller than the number, while multiples are equal to or larger.
Does every number have factors?
Yes, every positive integer has at least two factors: 1 and itself (except for the number 1, which has only one factor). Numbers with exactly two factors are called prime numbers.
How does the factor calculator handle large numbers?
The calculator uses a trial division algorithm up to the square root of the number. For very large numbers (e.g., over 15 digits), the calculation might take slightly longer but is processed efficiently in your browser to provide the full list of factors and prime components.