Prime Factorization Calculator
Enter an integer to find its prime factorization, repeated prime factors, exponential form, prime-number status, and positive divisor count.
Output
Prime Powers
Check
Prime Status
Count
Divisors
Live calculator
Prime factorization inputs
Break an integer into prime factors, exponential form, expanded factor form, primality status, and divisor count.
Prime factorization
2^3 x 3^2 x 5
Absolute value
360
Prime?
No
Divisors
24
Factor count
6
Prime factorization checks
| Measure | Value | Check |
|---|---|---|
| Expanded factors | 2 x 2 x 2 x 3 x 3 x 5 | The integer written as repeated prime factors. |
| Exponential form | 2^3 x 3^2 x 5 | Repeated prime factors grouped with exponents. |
| Prime number check | Not prime | A prime number has exactly two positive divisors: 1 and itself. |
| Positive divisor count | 24 | Multiply each prime exponent plus one to count positive divisors. |
Output
Prime Powers
Check
Prime Status
Count
Divisors
Break integers into prime factors
Expanded factor form
Show each repeated prime factor in multiplication order.
Exponent form
Group repeated factors such as 2 x 2 x 2 into 2^3.
Prime and divisor checks
Identify whether the value is prime and count positive divisors from exponents.
Prime factorization rule used on this page
Every integer greater than 1 can be written as a product of prime numbers, and that product is unique apart from order.
Working formulas
Prime factorization
n = p1^a x p2^b x p3^c ...
Each p is prime and each exponent counts repeated factors.
Divisor count
(a + 1)(b + 1)(c + 1) ...
Add one to each prime exponent and multiply the results.
Symbols
- p - prime factor
- A whole number greater than 1 whose only positive divisors are 1 and itself.
- a, b, c - exponents
- Counts of how many times each prime factor appears.
Interpret this calculation before using it
Prime factorization expresses an integer greater than one as a product of primes. The prime product is unique apart from factor order, so exponent form provides a compact identity that can be checked by multiplication. Divisor count follows from the exponents: add one to each exponent and multiply those counts.
Worked example
For 360, repeated division gives 360 = 2^3 x 3^2 x 5. Multiplying 8 x 9 x 5 returns 360. The number of positive divisors is (3 + 1)(2 + 1)(1 + 1) = 24 because a divisor can choose zero through three factors of 2, zero through two factors of 3, and zero or one factor of 5.
Assumptions to keep
- Input is interpreted as a whole number within the calculator's supported safe range.
- For a negative integer, factorization applies to its absolute value with a separate factor of -1.
- The values 0 and 1 do not have ordinary prime factorizations, so they require special result states.
Limits of this result
Trial division is practical for the calculator's bounded educational inputs but is not a general-purpose method for very large cryptographic integers. A factorization result should multiply back to the original magnitude. Use the exponent form for divisor work and the expanded form when tracing individual prime divisions.
Useful next step: GCD and LCM Calculator — Compare prime exponents across multiple integers to find their greatest common divisor or least common multiple.
Factorization with exponent and divisor checks
Formula steps that can be audited
- Scientific notation shows coefficient, exponent, engineering notation, and e notation.
- Prime factorization keeps expanded and exponent forms visible for checking.
- GCD and LCM outputs include factor rows so common and repeated factors can be inspected.
- Copy and print controls preserve the exact method labels used on the result cards.
Guardrails for integer and notation work
- Integer tools restrict inputs to whole numbers and explain special cases such as negative values.
- LCM calculations return a clear unavailable state if a result would exceed safe integer precision.
- Scientific notation separates ordinary decimal value, power of ten, and engineering multiples of three.
- Related links connect each calculator to the formula library and nearby math workflows.
Prime factorization support for number theory
Students
Check notation, factorization, and divisibility work while seeing each formula convention.
Teachers
Create examples for powers of ten, prime factors, common divisors, and common multiples.
Worksheet builders
Print results and method notes for answer keys, examples, and review sheets.
How it works in three quick steps.
Enter an integer
Use a whole number between negative one billion and one billion.
Read expanded factors
Review the repeated prime factors that multiply back to the absolute value.
Check exponent form
Use grouped prime powers for divisor counting, GCD, and LCM work.
Save or print factorization results
Copy result summary
Copy the final answer with formula labels into notes, documents, or worksheets.
Print the page
Print inputs, result cards, method tables, FAQs, and related math links.
Audit method tables
Use the supporting rows to verify powers of ten, factor powers, or divisibility logic.
Frequently Asked Questions
What is prime factorization?
How are negative integers handled?
Is 1 a prime number?
How does factorization help with GCD and LCM?
What does divisor count mean?
Related number theory tools
About this prime factorization calculator
Toolarithm's Prime Factorization Calculator is designed for divisibility work that needs more than a final answer. It shows expanded prime factors, grouped exponent form, whether the input is prime, and the positive divisor count. Negative inputs are handled with a visible -1 factor so the multiplication statement remains accurate.
The calculator supports classroom number theory, fraction reduction, simplifying radicals, GCD and LCM work, and worksheet answer keys. Prime factorization is also a bridge into modular arithmetic, cryptography concepts, and algebraic simplification. Keeping exponent form and divisor count visible makes the output useful for more than a quick lookup.
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 methodologyDates and sources
Review dates change only after a substantive method or content check.
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