Prime Factorization Calculator

Divide out the smallest prime that fits, again and again, until nothing is left. 360 gives 2 three times, 3 twice and 5 once, so 360 = 2^3 × 3^2 × 5. Every whole number above 1 has exactly one such decomposition.

Every whole number above 1 is either prime or a product of primes, and that product is unique apart from the order of its factors. This is the fundamental theorem of arithmetic, and it is why a factorization is an answer rather than one answer among several: 360 is 2 × 2 × 2 × 3 × 3 × 5 and nothing else. Writing it in exponent form keeps it short. Three twos, two threes and one five become 2^3 × 3^2 × 5, where the exponent counts how often that prime divides the number. Finding the factors is trial division. Take out every 2 you can, then every 3, then 5, 7 and upwards through the odd numbers. You can stop as soon as the divisor squared is larger than what is left over: any remaining factor smaller than that would already have been removed, so whatever remains above 1 must itself be prime. That is why checking 97 costs only a handful of divisions — after 3, 5 and 7 the square already passes 97. The decomposition is the foundation for other work. The greatest common divisor is built from the primes two numbers share, the least common multiple from the higher exponent of each prime, and a fraction reduces to lowest terms exactly when numerator and denominator have no prime in common.