Prime Factorization Calculator

Find the prime factors, divisors, factor pairs, and number properties of any integer up to 10¹².

Positive Integer (up to 1,000,000,000,000)

Prime Factorization

360 = 2³ × 3² × 5

Number Properties

Totient φ(n)96

Divisor Sum1,170

ClassificationAbundant

All Divisors (24)

1234568910121518202430364045607290120180360

Factor Pairs (12)

1×360

2×180

3×120

4×90

5×72

6×60

8×45

9×40

10×36

12×30

15×24

18×20

Factorization Visualizer

How it worksThe tree recursively splits each composite number into its smallest prime factor (left) and the remaining quotient (right) until only primes remain.

What is Prime Factorization?

Prime factorization expresses any integer as a unique product of prime numbers — called the fundamental theorem of arithmetic. For example, 360 = 2³ × 3² × 5. This calculator uses trial division up to the square root of the input, which is exact and fast for values up to a trillion. From the prime factors it also derives the full divisor list and factor pairs.

Related tools: GCD & LCM Calculator, Prime Number Checker, and the Base Converter.

Built and maintained by Meet Shah · Last updated 1 August 2026

What this tool is used for

Breaking a number into primes for a maths exercise or a puzzle.
Finding a greatest common divisor by comparing two factorisations.
Checking whether a number is prime by seeing whether it factors at all.
Reducing a fraction by cancelling shared prime factors.
Understanding a number's divisors from its prime structure.

Frequently Asked Questions

Why is the factorisation unique?: The fundamental theorem of arithmetic: every integer above 1 has exactly one prime factorisation up to ordering. That uniqueness is why 1 is excluded from the primes — admitting it would allow 12 to be written as 2²×3, 1×2²×3, and infinitely many more.
How does trial division decide when to stop?: At the square root of the remaining value, because any factor above it must pair with one below it that would already have been found. That bound is what makes trial division practical at all — checking every candidate up to n would be hopeless.
Why is factorisation hard for large numbers?: Because the best known classical algorithms are sub-exponential but still infeasible at cryptographic sizes. A 2048-bit semiprime is beyond any current machine, and RSA's security rests entirely on that gap between multiplying and un-multiplying.
Would a quantum computer break it?: Shor's algorithm factors in polynomial time, so a sufficiently large fault-tolerant quantum computer would. None exists, and the practical response is already underway — post-quantum standards were finalised in 2024 precisely because encrypted data captured now could be decrypted later.
What is the factorisation used for day to day?: Reducing fractions, finding GCD and LCM, and simplifying radicals — all of which are the same operation underneath. It is also how you check whether two numbers are coprime, which decides whether a modular inverse exists.

Common errors and gotchas

Expecting trial division to handle very large numbers, which it cannot in reasonable time.
Forgetting that 1 is not prime, which breaks the uniqueness of the factorisation.
Assuming a number with no small factors is prime, which needs a proper primality test.
Reading a repeated factor as a mistake, when multiplicity is part of the answer.
Using a factorisation result for cryptographic reasoning, where the numbers involved are far larger.