Prime Factorization Calculator

n = p₁^a₁ × p₂^a₂ × … Trial division to find all prime factors with multiplicity.

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Result

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How to use this calculator

  • Enter positive integer.
  • Read prime factorization + divisor count.

About this calculator

Every integer >1 has a unique prime factorization (Fundamental Theorem of Arithmetic). 360 = 2³ × 3² × 5. The number of divisors is the product of (exponent+1) terms: 360 has (3+1)(2+1)(1+1) = 24 divisors. Trial division is fastest for n < 10^12; beyond that, Pollard rho or quadratic sieve. Cryptographic security (RSA) relies on factoring being hard for products of two large primes.

Frequently asked

Why is factorization unique?+
Fundamental Theorem of Arithmetic — every integer >1 has exactly one prime factorization (up to order). Proven by Euclid's lemma.
How many divisors does n have?+
If n = p₁^a₁ × p₂^a₂ × …, divisor count = (a₁+1)(a₂+1)… A perfect square has odd divisor count.
How big can I factor here?+
Up to ~10^12 in milliseconds. Beyond that, slows linearly with √n. Crypto-sized (300-digit) needs specialized tools.
Why trial division?+
Simple and fast for small numbers. Used in school. Pollard rho is faster for n > 10^15. General number field sieve for crypto.
GCD/LCM use this?+
Conceptually: GCD = product of mins of each prime exponent. LCM = product of maxes. Algorithmically, Euclidean is faster than factoring.

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