Prime Factorization Calculator
Decompose any supported positive integer into its unique prime factors, exponents, and verified prime powers.
Excel export is ready for the demonstration value.
Enter an integer
Required. Enter a whole number from 2 to 9,007,199,254,740,991. Plain digits or correctly grouped commas are accepted.
Factorization results
Verified: multiplying all listed prime powers returns 360,360 exactly.
Prime factor breakdown
| Prime factor | Exponent | Prime power |
|---|---|---|
| 2 | 3 | 8 |
| 3 | 2 | 9 |
| 5 | 1 | 5 |
| 7 | 1 | 7 |
| 11 | 1 | 11 |
| 13 | 1 | 13 |
Each row is a distinct prime. The exponent is the number of times that prime occurs; the prime power is primeexponent.
How to use the Prime Factorization Calculator
What this calculator does
This calculator expresses one positive integer as a product of prime numbers. It reports both exponential notation, such as 2³ × 3², and an expanded list in which repeated factors appear individually. It also classifies the input as prime or composite, counts distinct and repeated prime factors, calculates the number of positive divisors, identifies the smallest and largest prime factors, and verifies that the factor powers multiply back to the original integer. The result is an exact arithmetic identity within the supported range; it is not a probability, approximation, or recommendation.
When to use it
Use prime factorization when simplifying fractions, finding greatest common factors or least common multiples, checking whether a number is prime, studying divisibility, or preparing algebra and number-theory work. It is also useful for validating hand-built factor trees and for creating a clean spreadsheet record of a calculation.
How to calculate
- The calculator opens with the demonstration value 360,360, complete results, and an immediately available XLSX workbook.
- Replace the value in Positive integer. Results update live after every valid edit; no separate Calculate button is needed.
- Read Prime factorization for compact exponent notation and Expanded form for every repeated factor.
- Review the metric cards and the Prime factor breakdown table, then select Download Excel to export the current canonical values.
- Select Reset to clear the demonstration and all calculated content. Reset may disable Download Excel until a complete valid integer is entered again.
Input guide
Positive integer is the only required input. Enter decimal digits representing a whole number from 2 through 9,007,199,254,740,991. You may use no separators, as in 360360, or correctly grouped ASCII commas, as in 360,360. Decimal points, signs, spaces inside the number, scientific notation, and incorrectly grouped commas are rejected rather than silently reinterpreted. Increasing the input does not necessarily increase the number of prime factors: a large prime has only itself as a prime factor, while a smaller composite number may contain many repeated factors.
Output guide
Prime factorization is the unique product of distinct primes raised to exponents. Expanded form writes the same identity with repetitions. Number type is Prime when the input has exactly one prime factor with exponent 1; otherwise it is Composite. Distinct prime factors counts table rows, while Total prime factors adds all exponents. Positive divisors uses the exponent rule (a₁ + 1)(a₂ + 1)...; a prime therefore has exactly 2 positive divisors. Smallest prime factor and Largest prime factor mark the range of primes present. The verification line confirms exact reconstruction. In the table, Prime factor names the prime, Exponent gives its multiplicity, and Prime power gives the numeric value of that repeated group.
Worked example
The startup example is 360,360. Repeated division and primality checks produce 2³ × 3² × 5 × 7 × 11 × 13. The expanded form therefore contains nine factors: 2, 2, 2, 3, 3, 5, 7, 11, and 13. There are six distinct primes and nine prime factors with repetition. The divisor count is (3 + 1)(2 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1) = 192. Multiplying 8 × 9 × 5 × 7 × 11 × 13 returns 360,360, matching the first-open verification and workbook.
Learn more
The uniqueness of this decomposition follows from the fundamental theorem of arithmetic. For a concise mathematical definition and notation, see Wolfram MathWorld's prime factorization reference.
How prime factorization works
A prime number has exactly two positive divisors: 1 and itself. A composite integer can be split into smaller factors, and those factors can be split again until every remaining factor is prime. Different factor-tree branches may look different, but after the primes are sorted, the final prime powers are identical.
The calculator uses exact integer arithmetic, deterministic primality testing for the supported range, and a factor-finding method designed for large composite integers. It never converts the input through floating-point arithmetic before factorization, which protects digits near the upper supported limit.
Common interpretation mistakes
- Including 1 as a prime factor: 1 is neither prime nor composite and does not appear in prime factorizations.
- Confusing all factors with prime factors: 12 has six positive divisors, but its prime factors are only 2 and 3.
- Dropping repeated factors: 12 is 2² × 3, not merely 2 × 3.
- Reading exponent notation as multiplication: 2³ means 2 × 2 × 2, not 2 × 3.
For additional practice distinguishing primes from composites, the Khan Academy prime and composite review provides worked examples and checks.