Factor Calculator
Find all factors, factor pairs, and prime factorization of any positive integer instantly. Perfect for math homework, number theory, and divisibility problems.
Fast, accurate, and completely free. Get detailed factorization results with step-by-step explanations.
Enter any positive whole number
Why Use Our Factor Calculator?
Instant Results
Find all factors, factor pairs, and prime factorization in milliseconds. Efficient algorithm works even for large numbers.
100% Accurate
Uses proven mathematical algorithms to find all factors. Perfect for homework, exams, and professional work.
Learn & Understand
See factor pairs and prime factorization to understand number relationships. Great for learning number theory.
What are Factors?
Factors (also called divisors) of a number are positive integers that divide the number evenly without leaving a remainder. In other words, if A × B = N, then both A and B are factors of N.
For example, consider the number 24:
- 1 × 24 = 24, so 1 and 24 are factors
- 2 × 12 = 24, so 2 and 12 are factors
- 3 × 8 = 24, so 3 and 8 are factors
- 4 × 6 = 24, so 4 and 6 are factors
- All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Every number has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers. Numbers with more than two factors are called composite numbers.
How to Find Factors
Method 1: Division Method
- Start with 1 and the number itself (always factors)
- Test each integer from 2 up to the square root of the number
- If it divides evenly, both the divisor and quotient are factors
- List all factors found in ascending order
Example: Find factors of 36
√36 = 6, so test numbers 1 to 6
1 divides → 1 and 36
2 divides → 2 and 18
3 divides → 3 and 12
4 divides → 4 and 9
6 divides → 6 and 6
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Method 2: Prime Factorization
- Find the prime factorization of the number
- Use combinations of prime factors to generate all factors
- Include 1 and the number itself
Example: Factors of 12
Prime factorization: 12 = 2² × 3
Combinations: 1, 2, 3, 2×2=4, 2×3=6, 2²×3=12
Factors: 1, 2, 3, 4, 6, 12
Common Uses for Factors
📐 Simplifying Fractions
Find common factors to reduce fractions to lowest terms. The GCD (greatest common divisor) is the largest factor shared by numerator and denominator.
📦 Division Problems
Factors help divide items into equal groups. For example, 24 items can be arranged in groups of 1, 2, 3, 4, 6, 8, 12, or 24.
📐 Geometric Arrays
Factor pairs represent possible rectangular arrangements. A number with 24 objects can be arranged as 1×24, 2×12, 3×8, or 4×6 arrays.
🔒 Cryptography
Prime factorization is crucial in RSA encryption. The difficulty of factoring large numbers into primes provides security for modern cryptographic systems.
🔢 Number Theory
Understanding factors is fundamental to number theory. It helps identify prime numbers, perfect numbers, and study divisibility properties.
📅 Scheduling
Factors help determine possible time intervals and group sizes for recurring events, shift schedules, and periodic activities.
Frequently Asked Questions
What are factors of a number?
Factors (or divisors) of a number are positive integers that divide the number evenly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 evenly.
How do you find all factors of a number?
To find all factors, divide the number by every integer from 1 up to the square root of the number. If it divides evenly, both the divisor and the quotient are factors. This method is efficient because factors come in pairs.
What are factor pairs?
Factor pairs are two numbers that multiply together to give the original number. For example, the factor pairs of 12 are (1,12), (2,6), and (3,4) because 1×12=12, 2×6=12, and 3×4=12.
What is the difference between factors and prime factors?
Factors include all numbers that divide evenly into the original number, while prime factors are only the prime numbers that multiply together to make the original number. For example, 12 has factors 1,2,3,4,6,12 but prime factors are only 2,2,3.
What is a prime number?
A prime number is a number greater than 1 that has exactly two factors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7. Numbers with more than two factors are called composite numbers.
Does every number have 1 as a factor?
Yes, every positive integer has 1 and itself as factors. These are called trivial factors. Numbers greater than 1 may have additional factors called proper factors or non-trivial factors.
How are factors used in real life?
Factors are used in many practical applications: dividing items into equal groups, creating rectangular arrays, simplifying fractions, finding common denominators, solving scheduling problems, and understanding divisibility in cryptography and computer science.
Can negative numbers be factors?
Technically yes, but by convention, when we list factors we typically only list positive factors. For example, while -2 divides 12 evenly, we usually only list 2 as a factor of 12.