Prime Factorization Calculator with Steps

Calculator Input

Integer to factorize

Recommended range: -1,000,000,000,000 to 1,000,000,000,000.

Negative number handling

Step detail level

Graph type

Show summary metrics

Keep expanded factor form

Show Plotly graph

Formula Used

Prime factorization writes a whole number as a product of prime powers: n = p₁^a₁ × p₂^a₂ × ... × p_k^a_k.

This calculator applies repeated trial division. It starts with the smallest prime, 2, removes every factor of 2, then checks odd candidates 3, 5, 7, and so on until the remaining value becomes 1.

After exponents are known, the divisor count uses: d(n) = (a₁ + 1)(a₂ + 1)...(a_k + 1). For negative inputs, the sign can be shown separately as -1 × the factorization of the absolute value.

How to Use This Calculator

Enter any whole number in the input box.
Choose whether to keep a negative sign as -1 × factors or ignore the sign.
Select full steps for a classroom-style ladder or compact steps for a short summary.
Pick a graph style and enable the chart when you want a visual exponent comparison.
Press Factorize Number to show the result above the form.
Review the factor table, step list, summary metrics, and reconstruction check.
Use the CSV button for spreadsheet export or the PDF button for a printable report.

Example Data Table

Number	Prime Factorization	Distinct Primes	Total Prime Factors	Total Divisors
360	2³ × 3² × 5	3	6	24
999	3³ × 37	2	4	8
1024	2¹⁰	1	10	11
2310	2 × 3 × 5 × 7 × 11	5	5	32

Frequently Asked Questions

1) What numbers can this calculator factorize?

It accepts whole numbers within the stated range, including negatives. The factorization is computed on the absolute value, and a -1 factor is shown when you keep the sign.

2) Why are 0 and 1 special?

Zero has no finite prime factorization because every prime divides zero. One is a unit, not a prime, so it has no prime factors.

3) What does the steps table show?

Each row shows the current value, the prime divisor used, and the quotient produced. Reading downward recreates the full repeated-division method taught in class.

4) Why do exponents appear?

Exponents compress repeated primes. For example, 2 × 2 × 2 × 3 becomes 2³ × 3, which is shorter and easier to compare.

5) How is divisor count computed?

After factorization, multiply one more than each exponent. For 360 = 2³ × 3² × 5¹, the divisor count is (3+1)(2+1)(1+1) = 24.

6) Can I use negative numbers?

Yes. When sign retention is enabled, the result is written as -1 times the prime factorization of the absolute value.

7) Why is large input limited?

Trial division is clear and reliable, but extremely large integers can be slow in a single page tool. The limit keeps calculations practical.

8) What does the chart mean?

The chart plots prime bases against their exponents, helping you see whether a number is built from repeated small primes or many distinct ones.