Prime factorisation

Break a whole number into primes. There is exactly one answer, which makes this the rarest thing on this site: a calculation with nothing to disown. Nothing you type here leaves your browser — it is worked out on your own device, so there is nothing to send and nothing to log.

There is exactly one answer, and this is it

Every whole number above 1 is a product of primes in exactly one way. Not “usually”, not “up to rearrangement of the interesting cases” — one way, always. That is the fundamental theorem of arithmetic, and it is why this page has nothing to disown: there is no rounding, no convention and no choice. 360 is 2³ × 3² × 5 and it cannot be anything else.

1 is not prime, and it is not composite either

This trips everybody, and the reason is the theorem above. If 1 counted as prime then 6 would be 2 × 3, and 1 × 2 × 3, and 1 × 1 × 2 × 3, and the “exactly one way” would be false. So 1 is left out by definition, and it has no prime factorisation at all — not an empty one by accident, but by design.

Why the powers matter

Written as powers, the factorisation answers other questions immediately. A number that is p^a × q^b has exactly (a+1)(b+1) divisors, because each divisor picks a power of each prime independently. 360 = 2³ × 3² × 5¹ therefore has 4 × 3 × 2 = 24 divisors, and you never have to look for them.

How it is worked out

Trial division, which is more than enough here. Divide by 2 as often as it goes, then by 3, 5, 7 and every odd number after, stopping at the square root — because a factor above the square root would have had a partner below it, and that partner would already have been found. Whatever is left when the loop ends is itself prime.

What it will not do

It will not factorise numbers above 9,007,199,254,740,991, which is the largest whole number a browser can hold without losing digits. Past there the number you typed is not the number being worked on.
It will not be fast on a large semiprime. Trial division is honest and simple; factorising a 40-digit product of two primes is the thing modern cryptography is built on being hard.
It will not factorise fractions or negatives. The factorisation of −12 is the factorisation of 12 with a minus sign, so the sign is dropped and said so.