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Prime Factorization Calculator

Find the prime factorization of any positive integer. Express any number as a unique product of prime numbers raised to their respective powers.

Reviewed by Christopher FloiedPublished Updated

This free online prime factorization calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Must be 2 or greater

Results

Prime Factorization

2^3 × 3^2 × 5

Distinct Prime Factors

2, 3, 5

Total Number of Divisors

24

Sum of All Divisors

1170

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Prime Factorization Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Prime Factorization Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Prime Factorization Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Prime Factorization Calculator

Prime factorization is the process of expressing a positive integer greater than 1 as a product of prime numbers. The Fundamental Theorem of Arithmetic guarantees that every such integer has a unique prime factorization, up to the order of the factors. This makes prime factorization the cornerstone of number theory. It is used to compute GCF and LCM, simplify radicals, analyze divisibility, and forms the mathematical basis of RSA cryptography. The security of RSA relies on the computational difficulty of factoring very large semiprimes (products of two large primes). While trial division works for small numbers, sophisticated algorithms like Pollard's rho, the quadratic sieve, and the general number field sieve are used for large numbers. Prime factorization also reveals the structure of a number's divisors — the number of divisors, their sum, and their product can all be computed directly from the factorization.

The Math Behind It

The standard algorithm for prime factorization is trial division: divide the number by 2 as many times as possible, then by 3, then by 5, and so on through successive primes up to √n. The quotient at each step is divided repeatedly by the current prime until it is no longer divisible. The remaining quotient (if greater than 1) is itself prime. From the factorization n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, we can derive: the number of divisors τ(n) = (a₁+1)(a₂+1)...(aₖ+1); the sum of divisors σ(n) = product of (pᵢ^(aᵢ+1) − 1)/(pᵢ − 1) for each prime power; the Euler totient φ(n) = n × product of (1 − 1/pᵢ). A number is a perfect square if and only if all exponents in its factorization are even. The factorization connects to the Mobius function μ(n), which is 0 if any aᵢ > 1, otherwise (−1)^k. Understanding prime factorization enables efficient computation of modular exponentiation, discrete logarithms, and quadratic residues.

Formula Reference

Fundamental Theorem of Arithmetic

n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ

Variables: p₁ < p₂ < ... < pₖ are primes, aᵢ ≥ 1

Number of Divisors

τ(n) = (a₁+1)(a₂+1)...(aₖ+1)

Variables: aᵢ = exponents in prime factorization

Worked Examples

Example 1: Factoring 360

Find the prime factorization of 360

Step 1:360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45 (three 2s)
Step 2:45 ÷ 3 = 15, 15 ÷ 3 = 5 (two 3s)
Step 3:5 ÷ 5 = 1 (one 5)

360 = 2³ × 3² × 5

Example 2: Counting Divisors of 360

How many divisors does 360 have?

Step 1:360 = 2³ × 3² × 5¹
Step 2:Number of divisors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24

360 has 24 divisors

Common Mistakes & Tips

  • !Stopping the factorization too early — the remaining quotient may itself be a large prime.
  • !Including 1 as a prime factor — 1 is not prime.
  • !Confusing prime factorization with listing all divisors.
  • !Not dividing by the same prime repeatedly — 12 = 2² × 3, not 2 × 3.

Related Concepts

Prime Number

The building blocks used in prime factorization.

GCF

Found by taking minimum exponents from prime factorizations.

LCM

Found by taking maximum exponents from prime factorizations.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Factorial Calculator

GCF Calculator (Greatest Common Factor)

LCM Calculator (Least Common Multiple)

Perfect Cube Calculator

Prime Number Calculator

Frequently Asked Questions

Is prime factorization unique?

Yes. The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has exactly one prime factorization (ignoring the order of factors).

How hard is it to factor large numbers?

Factoring large numbers (hundreds of digits) is computationally very difficult. No known polynomial-time classical algorithm exists, which is why RSA encryption is secure.

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