Prime Factorization Calculator

Prime factorization is the process of breaking a whole number down into the specific set of prime numbers that multiply together to produce it. Every whole number greater than 1 has one, and only one, prime factorization — a result so important it is called the Fundamental Theorem of Arithmetic. No matter which order you break a number down in, you always end up with the exact same set of prime factors (and their exponents). This uniqueness is what makes prime factorization such a powerful and reliable tool across mathematics, from simplifying fractions to modern cryptography. To find the prime factorization of a number by hand, repeatedly divide it by the smallest prime number that divides it evenly, and keep dividing the result by that same prime until it no longer divides evenly, then move on to the next prime. Let's factor 360. Start with the smallest prime, 2: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45. Now 45 is odd, so 2 no longer divides it — we move to the next prime, 3: 45 ÷ 3 = 15, 15 ÷ 3 = 5. Now 5 is no longer divisible by 3, so we move to the next prime, 5: 5 ÷ 5 = 1. We've reached 1, so we're done. Along the way we divided by 2 three times and by 3 two times and by 5 once, so 360 = 2 × 2 × 2 × 3 × 3 × 5, which is more compactly written using exponents as 2³ × 3² × 5. That compact exponent notation is standard because repeated prime factors are extremely common, especially in "round" numbers with lots of factors like 360, 720, or 1,000. Writing out every repeated multiplication becomes unwieldy for larger numbers, so mathematicians write 2 × 2 × 2 as 2³ (read "two cubed" or "two to the third power"), where the small raised number, the exponent, tells you how many times that prime appears in the factorization. This calculator uses real superscript Unicode characters (like ² and ³) so the result reads exactly like it would in a textbook, rather than an awkward "2^3" typed with a caret. Prime factorization directly powers several other calculators on this site. The GCF and LCM calculators can, in principle, be computed by comparing the prime factorizations of two numbers (take the lowest shared exponent of each common prime for GCF, the highest exponent of every prime involved for LCM), though the Euclidean algorithm used by those tools is faster for large numbers. Simplifying a fraction to lowest terms is really just canceling out prime factors that the numerator and denominator share. Even the Prime Number Checker on this site is closely related: a number is prime precisely when its prime factorization contains only one prime, raised to the power of 1 — itself. A subtlety worth understanding is why the algorithm only needs to test candidate primes up to the square root of the remaining number at each step, exactly like the trial-division method used by the Prime Number Checker. Once you have divided out every small prime factor and the remaining value is still greater than 1, that remaining value must itself be prime — there is no need to keep testing further, because if it had any factor at all besides 1 and itself, that factor would have already been found by an earlier, smaller candidate prime. This is exactly why the factorization of a large prime number (like 9,973) terminates quickly: the algorithm tries dividing by 2, 3, 5, 7… up to about 99 (the square root of 9,973), finds nothing, and correctly concludes the number is entirely its own single prime factor. A common misconception is that only "special" numbers have interesting factorizations. In reality, every composite number has exactly one correct factorization, and finding it (rather than guessing or memorizing it) is precisely what this calculator automates.