Factoring Calculator
Online factoring calculator to find prime factorization of any positive integer. Enter a number and get instant results showing all prime factors with exponents. Perfect for math homework and number theory.
About Factoring Calculator
Enter any positive integer and instantly get its prime factorization with exponents. The calculator shows all prime factors, complete factorization (e.g., 24 = 2³ × 3), all factors of the number, and factor pairs. Perfect for simplifying fractions, finding GCD and LCM, solving math problems, and understanding number properties. Supports large numbers and provides results in seconds.
Prime Factorization Principles
1. Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.
2. Fundamental Theorem of Arithmetic
Every integer greater than 1 can be uniquely expressed as a product of prime numbers. This means the prime factorization of any number is unique (up to the order of factors).
3. Factorization Method
The most efficient method is trial division: start with the smallest prime (2), test divisibility, divide if possible, and repeat with the next prime until the quotient is 1.
4. Exponents
When the same prime factor appears multiple times, we use exponents to simplify the notation. For example, 8 = 2 × 2 × 2 = 2³, and 36 = 2 × 2 × 3 × 3 = 2² × 3².
5. Special Cases
- Prime numbers: A prime number's factorization is itself (e.g., 7 = 7)
- Powers of primes: Numbers like 16 = 2⁴ or 27 = 3³
- Perfect squares: Numbers like 36 = 2² × 3² = (2 × 3)² = 6²
- Perfect cubes: Numbers like 8 = 2³ or 64 = 4³ = (2²)³ = 2⁶
Factorization Examples
12 = 2² × 3
Small composite number
24 = 2³ × 3
Multiple of 8
36 = 2² × 3²
Perfect square
60 = 2² × 3 × 5
Common denominator
100 = 2² × 5²
Perfect square
144 = 2⁴ × 3²
12 squared
210 = 2 × 3 × 5 × 7
Product of first 4 primes
1000 = 2³ × 5³
Power of 10
17 = 17
Prime number
97 = 97
Prime number
