Prime Factorization Calculator
Decompose any positive integer into its prime factors with step-by-step explanation.
Enter a positive integer greater than 1 (max 1,000,000,000)
Prime factorization is the process of decomposing a composite number into a product of prime numbers. According to the Fundamental Theorem of Arithmetic, every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers (up to the order of the factors).
Algorithm (Trial Division):
- Start with the smallest prime number, 2.
- While the current prime divides the number evenly, divide and record the factor.
- Move to the next potential divisor and repeat.
- Continue until the quotient becomes 1 or the remaining number is prime.
Example:
360 = 2³ × 3² × 5 = 2 × 2 × 2 × 3 × 3 × 5
Applications:
- Cryptography: RSA encryption relies on the difficulty of factoring large numbers.
- GCD/LCM: Finding greatest common divisors and least common multiples.
- Simplifying fractions: Reducing fractions to lowest terms.
- Number theory: Studying divisibility, perfect numbers, and more.
Time Complexity:
The trial division algorithm runs in O(√n) time, making it efficient for numbers up to about 10¹² (one trillion). For larger numbers, more sophisticated algorithms like Pollard's rho or the quadratic sieve are used.
Frequently Asked Questions
What is prime factorization?
Prime factorization is the process of breaking down a composite number into the product of its prime factors. According to the Fundamental Theorem of Arithmetic, every integer greater than 1 can be uniquely expressed as a product of prime numbers, regardless of the order. For example, 60 = 2^2 x 3 x 5.
Why is prime factorization important in cryptography?
RSA encryption, one of the most widely used security protocols, relies on the fact that factoring large numbers (products of two large primes) is computationally difficult. While multiplying two 300-digit primes takes milliseconds, factoring their product would take billions of years with current computers. This asymmetry provides security for online banking, secure communications, and digital signatures.
How do I find the GCD or LCM using prime factorization?
For GCD (Greatest Common Divisor): multiply the common prime factors with their lowest exponents. For LCM (Least Common Multiple): multiply all prime factors with their highest exponents. Example: For 12 = 2^2 x 3 and 18 = 2 x 3^2, GCD = 2^1 x 3^1 = 6, and LCM = 2^2 x 3^2 = 36.
What is the trial division algorithm?
Trial division is the simplest prime factorization method. It works by dividing the number by successive potential divisors starting from 2. When a divisor divides evenly, record it and continue with the quotient. Only test divisors up to the square root of the remaining number, since any factor larger than that would pair with a smaller factor already tested.