Prime factorization in simple words
A prime number is a whole number bigger than 1 that only 1 and itself divide. The first few are 2, 3, 5, 7, 11 and 13.
Prime factorization means writing a number as primes multiplied together. It is like breaking a number into its smallest building blocks.
Example: to break up 60, divide by 2 to get 30, divide by 2 again to get 15, divide by 3 to get 5, and 5 is prime. So 60 = 2 × 2 × 3 × 5, which is written 2² × 3 × 5.
Every whole number above 1 breaks up into primes in only one way. The order does not matter: 2 × 3 × 2 × 5 is still the same set of primes.
Repeated division by primes
Divide by 2 until the result is odd, then test odd divisors up to the square root of the remaining value. Each successful divisor is prime because its smaller factors have already been removed. Any remainder above 1 after this process is prime.
Every integer greater than 1 has a unique prime factorization, apart from the order of its factors. Exponents collect repeated factors into a compact product.
Example: factor 3,780
3,780 ÷ 2 = 1,890; ÷ 2 = 945; ÷ 3 = 315; ÷ 3 = 105; ÷ 3 = 35; ÷ 5 = 7; ÷ 7 = 1. Therefore 3,780 = 2² × 3³ × 5 × 7.
There are four distinct primes and seven factors with repetition. The number of positive divisors is (2 + 1)(3 + 1)(1 + 1)(1 + 1) = 48.
Reading the result
A square-free number has no repeated prime factor. For example, 30 = 2 × 3 × 5 is square-free; 12 = 2² × 3 is not. An input that is already prime produces a one-step ladder.
Use integers from 2 through 10¹². The number 1 has no prime factors, and decimals or negative inputs are not accepted. Factorization runs in your browser; inputs near the upper limit may take longer.
Listing primes
The “First N primes” mode lists up to 10,000 primes using the sieve of Eratosthenes; the 10,000th prime is 104,729. The “Primes between a and b” mode runs a segmented sieve: it crosses out multiples of every prime up to √b inside the range, leaving only primes.
Ranges can end as high as 10¹² and span up to 1,000,000 numbers. For example, there are 25 primes below 100 and 21 primes between 100 and 200.