Prime Factorization Calculator

Why the square root bound works

Any composite number has a factor at or below its square root. If both factors exceeded it, their product would exceed the number itself.

So trial division can stop at √n, and whatever remains above 1 must be prime. That turns an O(n) search into O(√n) — the difference between a thousand steps and thirty-one for a million.

In practice

360 factors as 2³ × 3² × 5. The full list is 2, 2, 2, 3, 3, 5, and grouping the repeats is what produces the exponent form.

The divisor count follows straight from those exponents: add one to each and multiply. (3+1)(2+1)(1+1) = 24, and 360 does indeed have 24 divisors.

That works because a divisor is built by choosing how many of each prime to include — none, one, two or three of the 2s, and so on. The choices multiply.

Why 1 is not prime

It looks arbitrary until you consider unique factorisation. Every number above 1 factors into primes in exactly one way, and that uniqueness is the foundation of most of number theory.

If 1 counted as prime, that would collapse: 6 could be 2 × 3, or 1 × 2 × 3, or 1 × 1 × 2 × 3, without limit. Excluding 1 keeps the factorisation unique, which is worth more than the tidiness of including it.