Prime Factorization Generator

Every integer greater than 1 can be written in exactly one way as a product of primes — that is the fundamental theorem of arithmetic. 360 is two cubed times three squared times five, and no other combination of primes gives 360. That uniqueness is what makes factorisation more than a curiosity: the greatest common divisor, the least common multiple and the divisor count all fall out of it.

Type the number and the page returns the factorisation with exponents in reduced form. The method is trial division: divide by 2 while it works, then by 3, and so on up to the square root of what remains — whatever is left above that is necessarily prime. It is the simplest algorithm there is and handles everyday numbers comfortably.

The difficulty of factoring large numbers is no accident: it underpins much of the cryptography in use. RSA leans on the gap between multiplying two three-hundred-digit primes, which is instant, and recovering the factors from the product, which no known classical computer does in reasonable time. This page handles numbers up to the trillions; above that, trial division stops being practical and methods like Pollard's rho and the number field sieve take over.