Toolarithm

Math Calculator

Prime Factorization Calculator

Find prime factors. Verify the product.

Factor an integer

Exact primes. Verified products.

Signed whole-number decimal only; no rounding or truncation.

View results

Calculation and report options

A bounded split-iteration budget, shared with enabled batch inputs. Increase effort for an unfinished cofactor.

Worked examples

Signed input magnitudes must satisfy ∣n∣≤264−1|n|\leq2^{64}-1. Negative numbers retain a separate −1-1 sign factor. Zero and units have dedicated explanations. Complete divisor lists are generated only when their exact count is at most 5,000.

Complete prime factorization verified by exact reconstruction.

Prime factorization

360=23⋅32⋅5360=2^{3}\cdot3^{2}\cdot5

Complete prime factorization verified by exact reconstruction.

Signed input: composite. Absolute-value classification: composite.

Standard: 1,000,000 split iterations. Split iterations used: 00. The effort budget is shared with enabled batch rows.

Exact reconstruction and prime powers

360=23⋅32⋅5360=2^{3}\cdot3^{2}\cdot5

Every listed prime is checked within the deterministic input range. Reconstruction multiplies verified factors and any explicitly unresolved cofactor; reconstruction alone is not a proof that an unfinished cofactor is prime.

232^{3}

Verified prime with multiplicity 33.

323^{2}

Verified prime with multiplicity 22.

515^{1}

Verified prime with multiplicity 11.

Properties of the positive magnitude

These functions apply to the absolute value, including for a negative input. They are computed only after its factorization is complete.

Positive divisor count

τ(360)=24\tau(360)=24

Sum of positive divisors

σ(360)=1170\sigma(360)=1170

Euler totient

φ(360)=96\varphi(360)=96

Product of distinct prime factors

rad⁡(360)=30\operatorname{rad}(360)=30

Prime factors with repetition

Ω(360)=6\Omega(360)=6

Distinct prime factors

ω(360)=3\omega(360)=3
  • Magnitude is not a perfect square.
  • Magnitude is not a perfect cube.
  • Magnitude is not square-free.
  • Divisor-sum classification: abundant.

Step-by-step division ladder

This sorted division ladder verifies the discovered prime product. It is an educational reconstruction, not a transcript of the runtime splitting algorithm. The sign factor is kept separate. An incomplete ladder ends at the unresolved cofactor.

  1. Division 1

    360÷2=180360\div2=180

  2. Division 2

    180÷2=90180\div2=90

  3. Division 3

    90÷2=4590\div2=45

  4. Division 4

    45÷3=1545\div3=15

  5. Division 5

    15÷3=515\div3=5

  6. Division 6

    5÷5=15\div5=1

Positive divisors and factor pairs

Lists refer to the absolute value. The full generated list is retained in copy and CSV; print uses the visible page.

Page 1 of 1. Positive divisors: 24; positive factor pairs: 12.

Positive divisors, ascending

11223344556688991010121215151818202024243030363640404545606072729090120120180180360360
What Can You Create?

Turn an integer into a checkable prime-product record

Grouped and repeated prime factors

Read the same exact decomposition as prime powers or repeated factors. Multiplicity matters: finding a prime once does not account for every copy in a prime power. A separate sign factor keeps negative examples understandable without treating minus one as a prime or confusing the signed input with its positive magnitude.

Divisor and coprimality properties

Use a completed decomposition to count positive divisors, add them, calculate Euler's totient and identify the distinct-prime product. Square, cube and square-free checks apply to the magnitude. These properties are withheld when an unresolved cofactor remains, so a useful partial product does not become an unsupported complete arithmetic summary.

A reproducible single or batch report

Preserve repeated batch rows and their source order, and keep each row's completion status. A full report includes exact reconstruction and the educational division ladder alongside generated divisor lists. The on-screen list can stay short while copied and downloaded working retain the complete generated data needed to review a result later.

Prime factorization formulas and worked examples

For a positive magnitude m>1m>1, the fundamental theorem of arithmetic gives m=∏i=1rpiaim=\prod_{i=1}^{r}p_i^{a_i} with distinct primes pip_i and positive integer exponents aia_i. The prime order may change, but the multiplicities do not. For negative input, retain the separate sign factor −1-1. A unit uses an empty prime product.

A complete classroom decomposition

Divide out each repeated prime and collect equal factors. Multiplying the prime powers reconstructs the starting magnitude.

360=2⋅2⋅2⋅3⋅3⋅5=23⋅32⋅5360=2\cdot2\cdot2\cdot3\cdot3\cdot5=2^3\cdot3^2\cdot5

A negative input keeps its sign

Factor the positive magnitude and retain the sign separately. The sign factor is not counted among the positive prime factors.

−72=(−1)⋅23⋅32,∣−72∣=72-72=(-1)\cdot2^3\cdot3^2,\qquad|-72|=72

Count choices of prime exponents

Each positive divisor selects an exponent from zero through the corresponding exponent in the full factorization. Multiplying these choice counts gives the total.

τ(360)=(3+1)(2+1)(1+1)=24\tau(360)=(3+1)(2+1)(1+1)=24

Sum the positive divisors

A finite geometric sum for each prime power accounts for all possible divisor contributions. Multiplication combines the independent choices.

σ(360)=(1+2+4+8)(1+3+9)(1+5)=1170\sigma(360)=(1+2+4+8)(1+3+9)(1+5)=1170

For a complete positive factorization, τ(m)=∏i(ai+1)\tau(m)=\prod_i(a_i+1), σ(m)=∏i(piai+1−1)/(pi−1)\sigma(m)=\prod_i(p_i^{a_i+1}-1)/(p_i-1) and φ(m)=m∏p∣m(1−1/p)\varphi(m)=m\prod_{p\mid m}(1-1/p). Euler's totient counts the positive integers up to the magnitude that are coprime to it; it is not the total number of divisors. These positive-integer formulas are not used for zero.

φ(360)=360(1−12)(1−13)(1−15)=96\varphi(360)=360\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac15\right)=96

Primality is classified deterministically only for 0≤m<2640\leq m<2^{64}, using the bounded witness theorem cited below. Splitting work is separately limited. An unfinished identity such as m=(verified prime product)⋅cm=(\text{verified prime product})\cdot c still names the unresolved composite cofactor cc; exact multiplication is not a license to call that cofactor prime.

Why Users Love This Tool

Separate exact evidence from unfinished computation

Whole numbers stay whole

  • Text parsing preserves large adjacent integers and rejects decimal inputs instead of silently truncating them. The magnitude limit is explicit and matches the deterministic primality domain.
  • Each listed prime is checked, multiplicities are retained, and an independent product identity closes the decomposition. Derived full divisor properties require completion.
  • Zero, units and negative signs receive their own explanations. A positive prime magnitude never makes a negative signed input a prime number under the selected convention.

Work that remains inspectable

  • A background worker keeps expensive splitting separate from the interface. Pending, cancelled and failed jobs cannot leave an old answer available as though it belonged to the new input.
  • The division ladder is explicitly a reconstruction of discovered factors, rather than an invented account of the runtime algorithm. It ends at the unresolved cofactor when necessary.
  • Complete generated lists stay in copied and downloaded reports. The visible page count and enumeration limit distinguish browsing a subset from claiming an incomplete divisor list is exhaustive.
Perfect For

For integer arithmetic, classroom explanations and exact checks

Students

Connect repeated division with prime powers and compare prime, composite, unit and zero examples. Inspect how the exponents determine divisor choices, and keep the sign convention visible when a negative integer has a prime or composite magnitude.

Technical readers

Check bounded machine-word integers with exact arithmetic and explicit computational limits. A composite classification can remain useful before every factor is resolved, provided its unresolved cofactor is retained and unsupported full properties are not inferred from a partial product.

Teachers

Build explanations around repeated factors, perfect squares, factor pairs and the distinction between totient and divisor count. Native result content and supporting formulas are readable without JavaScript for the default example. Complete exports let an answer key retain more detail than a compact visual list.

How It Works

How it works in three quick steps.

1

Enter an integer without rounding it

Type a signed whole-number decimal without grouping commas, a fraction slash or an exponent expression. The input is kept as text until exact integer parsing, so a decimal is rejected instead of being truncated to a different question. Negative inputs keep their sign separate, and zero and units receive dedicated explanations rather than a fabricated list of primes.

2

Review verified factors and the reconstruction

Read the grouped prime powers or switch to repeated prime factors. Each listed prime is checked in the supported deterministic range, and the reconstruction verifies their product together with any unresolved cofactor. If bounded splitting stops early, increase the effort setting or retain the explicitly partial report. Cancellation and changing the input terminate the current background calculation instead of blocking the interface.

3

Explore divisors and keep a complete report

For a complete nonzero factorization, compare the positive-magnitude divisor count, sum, totient and other properties. Browse the generated divisor or factor-pair list using its page controls, or enable an independent batch. Copy the explanation or LaTeX, download complete CSV working, share the enabled inputs, or print the visible report. Exports retain generated lists beyond the currently displayed page.

Download & Print

Retain the exact factors, warnings and complete generated working

Copy the explanation or LaTeX

Copy summary keeps completion messages, exact reconstruction, factor properties and division working. Copy LaTeX supplies the selected typeset product source for a compatible editor. If the clipboard is unavailable, use the complete CSV report instead of retyping a large integer from its visual display.

Download a complete CSV report

The rectangular report stores generated divisors and factor pairs in separate rows, not merely the current visual page. Long integer cells carry a text marker to avoid spreadsheet rounding. It also keeps batch completion warnings and the selected work policy. It has descriptive headers and is therefore a retained report rather than the numeric source template expected by local import.

Print the current report or save PDF

Print the selected grouped or repeated form, visible division ladder, enabled batch and current divisor page. Browser page margins isolate the report from navigation and controls. Keep a CSV when you need all generated list pages, and retain the explicit unresolved-cofactor message whenever a result is partial.

FAQ

Frequently Asked Questions

Is a negative number such as negative seven prime?
Under the ordinary positive-integer convention used here, a negative integer is not a prime number. Its absolute value can still be prime, and a signed factorization keeps a separate minus-one sign factor. The calculator distinguishes the signed input from the classification and divisor properties of its positive magnitude. This prevents a prime magnitude from being reported as though the original negative input itself were prime.
Why do zero and one need special explanations?
One and negative one are units: neither is prime or composite, and neither has prime factors. Their positive magnitude has one positive divisor. Zero has no finite prime factorization because a finite product of nonzero primes cannot equal zero. Every positive integer divides zero, so presenting an empty list with a finite zero divisor count would be misleading. The calculator withholds the positive-integer arithmetic-function cards for zero.
How large an integer can this tool handle exactly?
Signed decimal inputs are supported through the largest unsigned sixty-four-bit magnitude. Parsing and arithmetic use arbitrary-precision integers within that explicit domain, so nearby large integers are not merged by native floating-point rounding. The deterministic primality guarantee applies only inside the stated range. Values outside it, decimals, scientific e notation and general expressions are rejected rather than silently rounded, truncated or tested under an unsupported primality claim.
Is primality testing the same as finding every prime factor?
No. Determining that a supported integer is composite can be much quicker than splitting it into all of its prime factors. This tool uses a bounded splitting policy after deterministic primality checks. A difficult composite can therefore have a correct composite classification while part of its factorization is still unresolved. The result names that cofactor explicitly and does not publish full divisor counts, sums or totients from an unfinished decomposition.
What does an exact reconstruction prove?
The reconstruction checks that the displayed prime powers and any explicitly unresolved cofactor multiply to the positive magnitude. This detects missing or duplicated multiplicities, but a product identity alone does not establish that each factor is prime. The listed prime factors also undergo the supported deterministic primality test. A partial result remains partial even when its reconstruction identity is exact, and its warning is retained in copies and CSV reports.
How are positive divisors and factor pairs listed?
A complete factorization determines the exact positive divisor count before enumeration. When that count is at most five thousand, the tool generates the full ascending divisor list and positive factor pairs, including the repeated middle pair of a perfect square only once. On-screen page controls shorten the current view, while exports retain the entire generated list. Above the enumeration limit, exact factorization and arithmetic properties remain available without presenting a truncated list as complete.
Why can I cancel, retry or increase factorization effort?
Large composites do not all require the same amount of splitting work. The calculation runs in a separately owned browser worker with a bounded iteration policy. Cancel stops that worker, changing numerical inputs replaces it, and retry starts a fresh job. The effort budget is shared with enabled batch inputs, so earlier rows can consume work that later composites need. A lower remaining budget never turns an unresolved cofactor into a reported prime.
What do sharing, local import, CSV and printing retain?
A shared link retains the input, selected effort and report options, including batch text only when the batch is enabled. Numeric local import preserves source order and rejects invalid or oversized data without replacing prior inputs. Complete copies and CSV contain verified division working and all generated divisors, even when a list page or step panel is hidden. Printing follows the current visible report and divisor page, and partial reports keep their unresolved-cofactor warning.
About This Tool

About this Prime Factorization Calculator

Toolarithm's Prime Factorization Calculator is designed to make an integer decomposition understandable and checkable. A tidy product can hide several different questions: whether the original input is prime, whether its positive magnitude is prime, whether every prime multiplicity has been found, and whether enough information exists to list all divisors. This page keeps those questions separate. It reads a bounded signed decimal integer exactly, preserves its sign convention and shows both the discovered factors and their reconstruction identity. An unfinished computation remains explicitly unfinished rather than acquiring the appearance of a certified complete answer.

The interface supports learning as well as numerical checking. Prime powers emphasize repeated multiplicity, while an expanded product connects those exponents with repeated division. The ladder supplies a readable verification of the discovered factor product; it does not pretend to reproduce the internal splitting algorithm. Once a nonzero factorization is complete, positive-magnitude properties and a bounded divisor list become available. Their labels distinguish a count of divisors, a sum of divisors and a count of coprime positive integers, which are related arithmetic functions but answer different questions.

Practical computation also needs an honest stopping rule. Large inputs run in a cancellable background worker, and the selected effort policy bounds splitting attempts across the enabled job. Sharing retains the source settings needed to repeat that job, while copied and downloaded reports preserve its completion status and exact working. Visual pagination is a browsing choice, not a deletion of generated data. The result is intended for educational and technical verification inside the stated machine-word domain; it does not claim unbounded integer factorization or allow a product check by itself to stand in for a primality test.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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