How prime factorization works
The fundamental theorem of arithmetic states that every whole number greater than 1 can be written as a product of prime numbers in exactly one way, aside from the order of the factors.
- n — the number being factored
- pᵢ — the i-th prime factor
- eᵢ — how many times pᵢ divides n
More detail
Why the factorization is unique
The fundamental theorem of arithmetic guarantees that no matter how you break a number down, you land on the same set of primes and exponents. 60 is always 2² × 3 × 5 — never 2 × 3 × 5² or any other combination — which is why prime factorization is a reliable building block for other math, like finding a GCD or LCM.
How the calculator finds the factors
The tool trial-divides n by 2, then 3, 4, 5, and so on, dividing out each prime as many times as it fits before moving to the next candidate. Whatever is left over after dividing out every factor up to its square root is itself prime, so it's added as the last factor.
Quick check. Test 2 first, then 3, 4, 5, and so on. A number that isn't divisible by anything up to its own square root is prime itself, so you can stop early.
Frequently asked questions
What is the prime factorization of 60?
60 = 2² × 3 × 5, found by dividing out 2 twice (60 → 30 → 15), then 3 once (15 → 5), leaving the prime 5. Enter 60 above to see this exact breakdown and its 3 distinct prime factors.
Why is prime factorization unique?
The fundamental theorem of arithmetic guarantees every whole number above 1 has exactly one set of prime factors (up to order) — that's why the same input always produces the same result here, unlike regular factor pairs, which can vary.
What's the difference between prime factors and all divisors?
Prime factors are only the prime numbers that multiply to n (e.g. 12 = 2² × 3, so its prime factors are 2 and 3). All divisors also include composite combinations like 4, 6, and 12 itself — this tool shows only the prime building blocks and their exponents.
Why can't 1 be prime-factorized?
The fundamental theorem of arithmetic applies to numbers greater than 1; 1 has no prime factors at all, so entering 1 (or 0, a negative number, or a decimal) shows "—" instead of a result.