Decompose any integer into its unique prime factor powers, generate factor trees, compute total divisors count τ(N), and calculate sum of divisors σ(N).
Click "Factorize" to compute prime factors.
A student simplifying square root radicals evaluates √360: factoring 360 into 2³ × 3² × 5¹ = 6² × 10 reveals that √360 = 6√10. A computer scientist computing greatest common divisors uses prime factorizations: GCD(360, 240) = 2³ × 3¹ = 24. A cryptography researcher analyzing RSA public keys factors a modulus N to find secret prime factors p and q.
Prime factorization is the process of breaking down a composite number into a product of prime numbers that equal the original number. Enshrined in the Fundamental Theorem of Arithmetic, every positive integer greater than 1 has a single, unique prime factor decomposition (up to the order of factors).
For example, 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5¹. This tool calculates exponential prime notation, builds factor division trees, computes total divisor counts τ(N), calculates the sum of divisors σ(N), and lists all positive divisors. The following sections explain division algorithms, divisor functions, and practical applications in number theory.
When an integer N is submitted, the engine executes trial division using prime candidates (2, 3, 5, 7, 11...) to extract prime factors until N reduces to 1.
1. Trial Division Factor Extraction:
Extract factor 2 while N % 2 = 0.
Extract factor 3 while N % 3 = 0.
Test odd divisors d = 5, 7, 11... while d × d ≤ N.
2. Canonical Exponential Form:
Express factors as N = p₁^(e₁) × p₂^(e₂) × ... × p_k^(e_k).
3. Total Divisor Count τ(N):
The number of positive divisors is calculated by adding 1 to each exponent and multiplying:
τ(N) = (e₁ + 1) × (e₂ + 1) × ... × (e_k + 1)
4. Sum of Divisors σ(N):
The sum of all positive divisors is given by:
σ(N) = ∏ [(p_i^(e_i + 1) - 1) / (p_i - 1)]
Simplifying radicals and surds. Extracting perfect squares, cubes, or higher powers from prime factorizations simplifies radical expressions (e.g. √(2³ × 3² × 5) = 2 × 3 × √10 = 6√10).
Finding Greatest Common Divisors and Least Common Multiples. GCD(a, b) takes the minimum exponent of each shared prime factor, while LCM(a, b) takes the maximum exponent.
Simplifying complex fractions. Factoring numerators and denominators into prime components identifies shared factors for instant cancellation to lowest terms.
Public-key cryptography and RSA decryption. The security of RSA encryption relies on the fact that while multiplying two 1000-digit prime numbers is instantaneous, recovering prime factors from their product takes supercomputers thousands of years.
Enter any integer between 2 and 10¹⁵ for instant trial division factorization.
Review the Divisor Count τ(N) and Sum of Divisors σ(N) cards to analyze perfect numbers (where σ(N) - N = N, such as 6 and 28).
For finding shared divisors between multiple numbers, use our GCD Calculator or LCM Calculator. For primality testing, use our Prime Number Checker.
The factorization engine executes client-side in JavaScript using trial division and double-precision arithmetic. Numbers up to 10¹⁵ factor in under 1 millisecond.
| Feature | This Tool | Hand Factor Tree | Scientific Calculator |
|---|---|---|---|
| Exponential Form | p₁^e₁ × p₂^e₂ | Manual multiplication | pFact key |
| Total Divisors τ(N) | Automatic formula | Manual counting | Not listed |
| Sum of Divisors σ(N) | Automatic formula | Manual summation | Not listed |
| Divisors List | Complete array printout | Manual list | Not listed |
| Privacy | Client-side browser | Paper | Local device |
| Cost | Free | Free | Hardware purchase |
A perfect number is a positive integer equal to the sum of its proper positive divisors (excluding itself). For example, 6 has proper divisors 1, 2, 3 and 1 + 2 + 3 = 6.
Prime factorization is defined for positive integers greater than 1. Negative integers are factored by factoring their absolute value and prepending a factor of -1.
For a prime factor pᵉ, there are e + 1 choice options for its exponent in any divisor: p⁰, p¹, p²... pᵉ.
Prime Number Checker — Tests whether a number is prime using trial division up to √n.
GCD Calculator — Finds greatest common divisors using the Euclidean algorithm.
LCM Calculator — Finds least common multiples for 2+ numbers.