Prime Factorization Calculator
Break any positive integer into its prime factors with exponent form and factor tree steps.
Inputs
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
Number
= 360
Step 1
360 ÷ 2 = 180
Step 2
180 ÷ 2 = 90
Step 3
90 ÷ 2 = 45
Step 4
45 ÷ 3 = 15
Step 5
15 ÷ 3 = 5
Step 6
5 ÷ 5 = 1
Prime factorisation
= 2^3 × 3^2 × 5
Expanded
= 2 × 2 × 2 × 3 × 3 × 5 = 360
How it works
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)
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.