What is prime factorization?

It is writing a number as a product of prime numbers. For 12 the prime factorization is 2^2 × 3, meaning 2 × 2 × 3.

Is the prime factorization of a number unique?

Yes. The fundamental theorem of arithmetic says every integer greater than 1 has exactly one prime factorization, apart from the order in which you write the factors.

What does it mean when a number has one prime factor?

If the only prime factor is the number itself (with exponent 1), the number is prime — for example 17 = 17.

Why can’t 1 be prime-factorized?

1 has no prime factors, and 1 is not itself prime, so it has no prime factorization. Prime factorization is defined for integers of 2 or more.

How is prime factorization used?

It is the basis for finding the greatest common factor and least common multiple, simplifying fractions, and much of number theory and cryptography.

Sumivo · Mathematics

Prime Factorization Calculator

Break a whole number into the product of its prime factors, with exponents — the unique prime factorization guaranteed by the fundamental theorem of arithmetic.

01 / inputs

Whole number (2 or greater)

integer

02 / result

PRIME FACTORIZATION

2^3 × 3^2 × 5

Factorization: 2^3 × 3^2 × 5
Distinct primes: 3
Prime?: No

Calculation trace

2^3 = 8
8
Factor out 2
3^2 = 9
9
Factor out 3
5
5
Remaining prime

Maintained by Jack Shi

How it works

Prime factorization writes a whole number as a product of prime numbers. The tool uses trial division: it divides by 2 as many times as possible, then by each odd number 3, 5, 7, … up to the square root of what remains, recording how many times each prime divides in. Any value left above 1 at the end is itself a prime factor. By the fundamental theorem of arithmetic, every integer greater than 1 has exactly one such factorization (apart from the order of the factors) — for example 360 = 2^3 × 3^2 × 5.

Assumptions & limits

Defined for integers greater than 1; 1, zero, negatives, and decimals are rejected.
Exponents show repeated primes, so 8 is written 2^3 rather than 2 × 2 × 2.
Inputs are capped at 1,000,000,000,000 so the factorization stays fast in the browser.

Verified runs

Synthetic examples executed at build revision 24533ff on 2026-10-29 UTC. Cards round decimal results to 6 significant figures; matching uses full precision and the tolerance below. Matches cover the displayed fields, not every possible input or clinical or engineering validation.

Inputs: Integer: 360 integer
Result: Prime factorization: 2^3 × 3^2 × 5 prime powers (expected Prime factorization: 2^3 × 3^2 × 5 prime powers) ✓
Inputs: Integer: 2 integer
Result: Prime factorization: 2 prime powers (expected Prime factorization: 2 prime powers) ✓
Inputs: Integer: 100 integer
Result: Prime factorization: 2^2 × 5^2 prime powers (expected Prime factorization: 2^2 × 5^2 prime powers) ✓

How these runs were checked

Method: prime-factorization@1.0.0. Expectations come from the listed test fixtures or hand arithmetic; they are not copied from these executions. Omitted optional inputs use the method defaults. Run dates record execution, not expert review or fresh measurements.

Download all verified runs (JSON) — full inputs, returned values, expectations, timestamps and buildRevision.

normal: 2026-10-29T06:04:06.920Z. Absolute tolerance: 1e-10. Text and booleans match exactly. Full-precision result: Prime factorization: 2^3 × 3^2 × 5 prime powers. Expected: Prime factorization: 2^3 × 3^2 × 5 prime powers. Expectation: packages/calculator-core/tests/prime-factorization.test.ts
boundary: 2026-10-29T06:04:06.920Z. Absolute tolerance: 1e-10. Text and booleans match exactly. Full-precision result: Prime factorization: 2 prime powers. Expected: Prime factorization: 2 prime powers. Expectation: packages/calculator-core/tests/prime-factorization.test.ts
sensitivity: 2026-10-29T06:04:06.920Z. Absolute tolerance: 1e-10. Text and booleans match exactly. Full-precision result: Prime factorization: 2^2 × 5^2 prime powers. Expected: Prime factorization: 2^2 × 5^2 prime powers. Expectation: packages/calculator-core/tests/prime-factorization.test.ts

Reproduce from the source checkout with pnpm --filter @sumivo/web gen:verified-runs. A rebuild changes execution times; the revision identifies its Git starting point, including any uncommitted changes made for that build.

FAQ

What is prime factorization?: It is writing a number as a product of prime numbers. For 12 the prime factorization is 2^2 × 3, meaning 2 × 2 × 3.
Is the prime factorization of a number unique?: Yes. The fundamental theorem of arithmetic says every integer greater than 1 has exactly one prime factorization, apart from the order in which you write the factors.
What does it mean when a number has one prime factor?: If the only prime factor is the number itself (with exponent 1), the number is prime — for example 17 = 17.
Why can’t 1 be prime-factorized?: 1 has no prime factors, and 1 is not itself prime, so it has no prime factorization. Prime factorization is defined for integers of 2 or more.
How is prime factorization used?: It is the basis for finding the greatest common factor and least common multiple, simplifying fractions, and much of number theory and cryptography.

Related calculators

SOURCES

OpenStax, Rice University
Prealgebra 2e ↗
Prime factorization and the fundamental theorem of arithmetic
Accessed 2026-09-02