Prime Factorization Calculator
Break any number into its prime factors with a factor tree.
About Prime Factorization Calculator
The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorisation (up to ordering). For example, 360 = 2³ × 3² × 5. The factorisation is found by trial division: divide by 2 repeatedly until odd, then try 3, 5, 7, … up to √n. Any remaining factor greater than 1 must itself be prime.
Formulas
Fundamental Theorem of Arithmetic
n = p₁^e₁ × p₂^e₂ × … × pₖ^eₖ (each pᵢ prime, unique up to order)
Number of divisors from factorisation
τ(n) = (e₁+1)(e₂+1)…(eₖ+1)
Worked example
With the default values below, Prime Factorization Calculator returns Prime Factorization of 2^3 × 3^2 × 5. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Numberaccepts 2–1,000,000,000
- 360
Results
- Prime Factorization
- 2^3 × 3^2 × 5
- Expanded Form
- 2 × 2 × 2 × 3 × 3 × 5
- Is Prime?
- No — composite
- Total Prime Factors (with multiplicity)
- 6
- Factor Tree Steps
- 360 ÷ 2 = 180; 180 ÷ 2 = 90; 90 ÷ 2 = 45; 45 ÷ 3 = 15; 15 ÷ 3 = 5; 5 ÷ 5 = 1
Step by step
- 1Number= 360
- 2Step 1
- 3Step 2
- 4Step 3
- 5Step 4
- 6Step 5
- 7Step 6
- 8Prime factorisation= 2^3 × 3^2 × 5
- 9Expanded= 2 × 2 × 2 × 3 × 3 × 5 = 360
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the prime factorisation of 360?
360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45. Then 45 ÷ 3 = 15, ÷ 3 = 5. So 360 = 2³ × 3² × 5. It has (3+1)(2+1)(1+1) = 24 factors.
Why is prime factorisation unique?
The Fundamental Theorem of Arithmetic guarantees it: every integer > 1 is either prime or can be expressed as a product of primes in exactly one way, apart from the order of the factors.
What is the largest number I can factorise here?
Up to 1 000 000 000 (10⁹). Trial division to √(10⁹) ≈ 31 623 is fast in a browser. For larger numbers, more sophisticated algorithms like Pollard's rho are needed.