The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers. This calculator finds that unique factorization using trial division, showing each step and listing all divisors of the number.
How to Use
Enter a positive integer to see:
- Prime factorization: The number expressed as a product of primes
- All divisors: Complete list of numbers that divide evenly
- Steps: Trial division process
Trial Division Algorithm
Divide by 2 repeatedly, then try odd numbers from 3 up to √n. Each time a divisor is found, divide it out completely before moving to the next candidate. Any remaining value > 1 is itself prime.
Applications
- GCD/LCM: Found via prime factorizations
- Simplifying fractions: Cancel common prime factors
- Cryptography: RSA relies on difficulty of factoring large numbers
- Divisor counting: If n = p₁^a₁ × p₂^a₂ × ..., divisor count = (a₁+1)(a₂+1)...
Step-by-Step Instructions
- 1Enter a positive integer.
- 2View the prime factorization in standard form.
- 3See all divisors of the number.
- 4Study the step-by-step trial division.
- 5Check the total divisor count.