Prime Factorization Calculator
Find the prime factorization of any integer up to 12 digits — exponent form like 360 = 2³ × 3² × 5, the full factor list, and a prime check. Free.
About the Prime Factorization Calculator
Every integer greater than 1 breaks down into a unique product of prime numbers — the fundamental theorem of arithmetic. This prime factorization calculator finds that breakdown instantly: enter a number like 360 and you get the exponent form 2^3 × 3^2 × 5, the expanded list 2 × 2 × 2 × 3 × 3 × 5, and a table of each prime with its exponent.
The calculator uses trial division — dividing out 2s, then 3s, then testing candidates of the form 6k ± 1 up to the square root — which handles any integer up to 12 digits in a fraction of a second, right in your browser. If the number has no factors below its square root, it is itself prime, and the tool tells you so.
Students use prime factorization to simplify fractions, find LCM and HCF, and reduce radicals; it is a staple of middle-school and O-level math across Pakistan, India and worldwide. Programmers and puzzle solvers use it for divisor counting and number-theory problems.
How to Use the Prime Factorization Calculator
- 1Enter a whole number of 2 or more (up to 12 digits).
- 2Read the exponent form and the expanded prime factor list instantly.
- 3Check the prime → exponent table for the structured breakdown.
- 4If the number is prime, the tool tells you directly.
Frequently Asked Questions
What is the prime factorization of 360?
360 = 2 × 2 × 2 × 3 × 3 × 5, written compactly as 2^3 × 3^2 × 5. You can verify: 8 × 9 × 5 = 360. Enter 360 in the calculator to see the same breakdown with the factor table.
How do I find prime factors by hand?
Divide by the smallest prime that fits, repeatedly. For 84: divide by 2 to get 42, by 2 again to get 21, by 3 to get 7, and 7 is prime — so 84 = 2^2 × 3 × 7. A factor tree draws the same process as branching pairs.
How is prime factorization used to find LCM and HCF?
Factor both numbers, then: HCF takes each common prime at its lowest power, LCM takes every prime at its highest power. For 12 = 2^2 × 3 and 18 = 2 × 3^2: HCF = 2 × 3 = 6 and LCM = 2^2 × 3^2 = 36.
What happens if I enter a prime number like 97?
The factorization is just the number itself — 97 = 97 — and the calculator explicitly labels it prime. A prime has exactly two divisors, 1 and itself, so it cannot be broken down further.
Why is there a 12-digit limit?
Trial division checks candidates up to the square root of the number, so a 12-digit input needs up to about a million checks — still instant in a browser. Much larger numbers require advanced algorithms, and JavaScript's safe-integer range also ends near 16 digits.
Is 1 a prime number, and what is its factorization?
No — 1 is neither prime nor composite, and it has no prime factorization (an empty product). That convention keeps factorizations unique: if 1 were prime, 6 could be 2 × 3 or 1 × 2 × 3 or 1 × 1 × 2 × 3, breaking the fundamental theorem of arithmetic.