Prime Factorization Calculator
| Famous n | Prime factorization | Divisors |
|---|---|---|
| 60 | 22 × 3 × 5 | 12 |
| 84 | 22 × 3 × 7 | 12 |
| 360 | 23 × 32 × 5 | 24 |
| 997 | prime | 2 |
| 1,024 | 210 | 11 |
| 1,000,000 | 26 × 56 | 49 |
| 123,456 | 26 × 3 × 643 | 28 |
| 999,999 | 33 × 7 × 11 × 13 × 37 | 64 |
Every whole number above 1 is either a prime or a product of primes, and that product is unique - the Fundamental Theorem of Arithmetic is the reason factor trees always land in the same place no matter how you climb them. This calculator does the climb for you: type a number up to 10 trillion and read its prime factorization with exponents, plus the divisor count that falls out of it for free.
The divisor bonus is the part homework never mentions: if n = 2² × 3 × 7, then n has (2+1)(1+1)(1+1) = 12 divisors - you never have to list them. The exponent+1 product rule turns any factorization into an instant count, which is why factorization is the master key for GCF, LCM, fraction simplifying and every divisibility argument.
The honest boundary: trial division by a 2-3-5 wheel settles anything up to 10 trillion in well under a second, because the worst case - a prime - only needs checks up to its square root. Cryptographic sizes (hundreds of digits) are hard precisely because no calculator on Earth can factor them quickly; that difficulty is what keeps your bank account safe.
How to use
- Type the number - integers up to 10 trillion; the checker floors decimals so 84.0 works fine.
- Read the factorization with exponents - 84 = 2² × 3 × 7 means two 2s, one 3, one 7.
- Use the divisor count for homework shortcuts - divisibility, fraction simplifying and GCF/LCM all fall out of the exponent pattern.
- Try the famous row in the table - 997 (prime), 1,000,000 = 2⁶ × 5⁶, 999,999 = 3³ × 7 × 11 × 13 × 37 - to see how wildly divisor counts swing.
Frequently asked questions
What is a prime factorization?
Writing a number as a product of primes with exponents: 84 = 2² × 3 × 7. The Fundamental Theorem of Arithmetic guarantees this form is unique - factor 84 by any route (factor tree, division ladder, guesswork) and you always land on the same four primes.
How do I find prime factors by hand?
Divide by 2 as long as it goes, then 3, then 5, then 7 - always in prime order, rebuilding the number on paper: 84 ÷ 2 = 42 ÷ 2 = 21 ÷ 3 = 7. Stop when the remaining number is prime or 1. You never need to test a factor past the square root of what remains, which is why hand factoring small numbers is quick.
Why does the number of divisors come from the exponents?
Each divisor picks how much of each prime to take: for 2² you can take 2⁰, 2¹ or 2² - three choices. Multiply the choices: (2+1)(1+1)(1+1) = 12 divisors for 84. The (exponent+1) product rule is the entire formula, and it works on any factorization.
What is prime factorization used for?
GCF and LCM (take the minimum or maximum exponent of each prime), simplifying fractions to lowest terms, checking divisibility at a glance, and cryptography - RSA encryption is built on the fact that multiplying two large primes is instant while factoring the product back is effectively impossible.
Why does the calculator stop at 10 trillion?
Trial division checks primes up to the square root - about 3.16 million here, which runs in a blink. Numbers with hundreds of digits (the RSA range) are not slow-by-degree but practically impossible to factor; honest tools state the wall instead of pretending. For schoolwork, puzzles and everyday checking, 10 trillion is beyond generous.