Prime Factorization Calculator
Enter any integer from 2 up to 1,000,000,000,000 and the calculator instantly returns its prime factorisation (e.g. 360 = 2³ × 3² × 5), tells you whether it is prime or composite, and lists τ(n) the number of divisors, σ(n) the sum of divisors and every positive divisor — handy for simplifying fractions, GCD/LCM, number-theory homework and RSA practice.
Please enter a whole number between 2 and 1,000,000,000,000.
Prime factorisation
2³ × 3² × 5
2 × 2 × 2 × 3 × 3 × 5
Number of divisors τ(n)
24
τ(n) = Π (eᵢ + 1), counts 1 and n.
Sum of divisors σ(n)
1,170
σ(n) = Π (pᵢ^(eᵢ+1) − 1) ÷ (pᵢ − 1).
Per-factor breakdown
| Prime | Exponent | Contribution |
|---|---|---|
| 2 | 3 | 8 |
| 3 | 2 | 9 |
| 5 | 1 | 5 |
All positive divisors
Uses 6k±1 trial division — factorises any integer up to 10¹² instantly.
Formula
n = p₁^e₁ × p₂^e₂ × … × p_k^e_k | τ(n) = Π(eᵢ + 1) | σ(n) = Π (pᵢ^(eᵢ+1) − 1) ÷ (pᵢ − 1)
- · By the Fundamental Theorem of Arithmetic, every integer > 1 has a unique prime factorisation (order ignored).
- · A prime number has exactly two positive divisors: 1 and itself; 2 is the smallest and the only even prime.
- · Uses 6k ± 1 trial division: divide out 2 and 3, then test 5, 7, 11, 13, … up to √n; anything left over is automatically prime.
- · With n capped at 10¹², trial division only needs to reach √n ≈ 10⁶, fast enough to run instantly in the browser.
- · The divisor count τ(n) is the product of (eᵢ + 1); the divisor sum σ(n) uses the geometric-series formula, so you do not need to enumerate divisors first.
- · A perfect number satisfies σ(n) = 2n (e.g. 6, 28, 496, 8128) — you can verify any of these in the tool.
- · When the divisor count exceeds 256 (e.g. highly composite numbers) the divisor list is suppressed for layout, but τ(n) and σ(n) are still computed exactly.
Frequently asked
Why is the upper limit 10¹² (one trillion)?
JavaScript Number can hold integers up to 2⁵³ − 1 ≈ 9 × 10¹⁵ precisely, but trial division has to walk up to √n. If n is much larger, the worst case (n a product of two primes near √n) requires testing 10⁸+ odd candidates, which stalls in a phone browser. Capping at 10¹² keeps √n around 10⁶, giving instant answers in every case. For very large numbers use Wolfram Alpha, SageMath, or algorithms like Pollard rho / ECM.
How can I use prime factorisation to find GCD and LCM?
Factor each number: a = Π pᵢ^aᵢ, b = Π pᵢ^bᵢ. Then GCD = Π pᵢ^min(aᵢ, bᵢ) and LCM = Π pᵢ^max(aᵢ, bᵢ). The identity GCD(a, b) × LCM(a, b) = a × b always holds. Example: 60 = 2² × 3 × 5, 126 = 2 × 3² × 7, so GCD = 2 × 3 = 6 and LCM = 2² × 3² × 5 × 7 = 1260.
Can I use this to simplify fractions?
Yes. Factor numerator and denominator, then cancel any shared prime factors (the GCD). Example: 84/180 with 84 = 2² × 3 × 7 and 180 = 2² × 3² × 5 share 2² × 3 = 12, so 84/180 simplifies to 7/15.
How does this relate to RSA encryption?
RSA security rests on the fact that multiplying two huge primes is easy, but recovering them from the product is computationally infeasible. Real-world RSA moduli are 2048–4096 bits (far beyond 10¹²), so no browser-based trial division can break them. This tool is for teaching and small examples only, not for cracking RSA keys.
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