Prime Factorization Calculator

Every number carries a secret structure: a unique set of primes that multiply into it. Factor 84 and you find 2^2 x 3 x 7; factor anything else and you find its own unique fingerprint. That fingerprint decides the divisors, the GCD and LCM with other numbers, whether a fraction simplifies - and it is why cryptography exists. But finding it by hand means testing primes one by one, and getting to the square root of a large number by hand is a miserable slog.

Prime Factorization Calculator does it instantly and exactly. Trial division with a 6k+1 wheel factors numbers up to 100 trillion in milliseconds, reporting every factor with its exponent. The divisor count and divisor sum follow automatically, and prime numbers are announced as such. Paste one number or a whole list - every factorization lands in its own block.

• Exact factorization: Every factor and exponent, to the last digit.
• Up to 100 trillion: Large numbers factor in milliseconds.
• Fast wheel method: 6k+1 trial division skips a third of all candidates.
• Divisor count and sum: Derived straight from the exponents.
• Prime detection: Primes are recognized and announced.
• Batch input: Factorize a whole list at once.
• Copy or download: Results ready for notes and worksheets.
• Free and private: Everything runs in your browser; nothing is uploaded.

1. Enter numbers - one per line, or separated by spaces or commas.
2. Read each factorization with its exponents, in its own block.
3. Check the divisor count and divisor sum for each number.
4. Note the primes - numbers with no factors beyond themselves are flagged.
5. Verify by multiplying the factors together - the product line confirms the answer.
6. Copy or download the results.

Example 1 - The classic. Enter 84: 2^2 x 3 x 7, with 12 divisors and sum 224.

Example 2 - A perfect square. Enter 144: 2^4 x 3^2. The even exponents reveal the square, and the divisor count (4+1)(2+1) = 15 - an odd count, which is the signature of a perfect square.

Example 3 - A prime. Enter 97: the tool reports it as prime, having tried the square root of candidates and found none.

Example 4 - A product of two primes. Enter 3233: 53 x 61 - a tiny RSA key, factored in milliseconds, which is exactly why real keys are hundreds of digits long.

Example 5 - Batch. Enter 12, 30, 210 and read three factorizations in one run.

• Instant and exact: No hand testing of primes.
• Deeper insight free: Divisor count and sum included.
• Handles large numbers: Up to 10^14 in milliseconds.
• Batch friendly: Whole lists in one pass.
• Verifiable: The product line checks every answer.
• Free and private: Runs entirely in the browser with nothing uploaded.