Prime Factorization Calculator with Steps

Prime breakdown made practical for students, engineers, and enthusiasts with detailed guidance. Follow every division, validate factors, and compute advanced number properties with ease. Get canonical forms, exponent tables, phi, sigma, and insights in seconds. Save, share, export CSV and PDF reports easily anywhere.

Digits only. Up to typical 64-bit range for speed.
Clear
Try 84 Try 360 Try 2310 Try 9973
Formula used

We factor using optimized trial division. Remove factors of 2 and 3, then test divisors of the form 6k±1 up to √n. Each successful division records a step and reduces n.

  • Canonical form: n = ∏ p_i^{e_i}.
  • Divisors count: τ(n) = ∏ (e_i + 1).
  • Sum of divisors: σ(n) = ∏ (p_i^{e_i+1}-1)/(p_i-1).
  • Euler’s totient: φ(n) = n ∏ (1 - 1/p_i).

For very large inputs, trial division may be slow. This tool prioritizes clarity of steps.

How to use
  1. Enter a positive integer and click Factorize.
  2. Review the canonical factorization and exponents table.
  3. Inspect the step-by-step divisions and optional factor tree.
  4. Export results as CSV or a PDF report for records.
  5. Copy a shareable URL so others can view the same input.

Example inputs below help you verify expected outputs and formats.

Example data
nCanonical factorizationτ(n)σ(n)φ(n)
842^2 × 3 × 71222424
3602^3 × 3^2 × 524117096
23102 × 3 × 5 × 7 × 11326912480
99739973299749972
Common Numbers and Factorizations
nFactorization
122^2 × 3
182 × 3^2
242^3 × 3
302 × 3 × 5
362^2 × 3^2
482^4 × 3
602^2 × 3 × 5
722^3 × 3^2
902 × 3^2 × 5
1002^2 × 5^2
1202^3 × 3 × 5
1442^4 × 3^2
1802^2 × 3^2 × 5
2102 × 3 × 5 × 7
Prime Powers (pe) Quick Table
p \ e 1 2 3 4 5 6
2248163264
3392781243729
5525125625312515625
7749343240116807117649
11111211331146411610511771561
13131692197285613712934826809
Arithmetic Functions Examples
nFactorizationτ(n)σ(n)φ(n)
362^2 × 3^299112
602^2 × 3 × 51216816
842^2 × 3 × 71222424
902 × 3^2 × 51223424
2102 × 3 × 5 × 71657648
23102 × 3 × 5 × 7 × 11326912480
First 25 Primes
2357111317192329313741434753596167717379838997
FAQs

Any positive integer that fits typical 64-bit range. Extremely large inputs may take longer with trial division.

We record only successful divisions for clarity. Attempting all non-divisors would clutter the steps and slow rendering.

We apply standard multiplicative formulas using the prime powers from the factorization. See the Formula section for exact expressions.

No. By definition, primes are integers greater than 1 with exactly two positive divisors.

Yes. Use CSV buttons for spreadsheets and the PDF button for printable reports. The Share button copies the current URL with the input embedded.

Trial division grows with √n. For very large semiprimes, advanced algorithms help, but they don’t produce straightforward human-readable steps.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.