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.