Prime powers, expanded factors, and divisor count
The power form groups repeated primes; the expanded form lists each occurrence. The divisor count includes 1 and the original number. A number with one factor of exponent one is prime; other accepted numbers are composite.
To compare the common prime factors of two integers, use the GCD and LCM Calculator .
The formula
Trial division removes all factors of 2, then tests odd candidates up to the square root of the remaining number. Any remainder above 1 is prime. Divisor counting chooses an exponent from zero through each factor’s exponent, then multiplies the number of choices.
Worked example
360 = 2³ × 3² × 5. The expanded product is 2 × 2 × 2 × 3 × 3 × 5. There are three distinct primes and (3 + 1)(2 + 1)(1 + 1) = 24 positive divisors.
How to use this calculator
- Enter an integer of at least 2 within the supported range.
- Calculate and compare the grouped powers with the expanded product.
- Expand the factor table to see each prime and its exponent.
Prime factors versus all factors
The factors of 12 include 1, 2, 3, 4, 6, and 12. Its prime factorization is 2² × 3. Numbers such as 4 and 6 are factors but are not prime, so they do not appear in the prime-factor table. The displayed divisor count counts all positive divisors, rather than only the primes.
How repeated primes help
A grouped representation makes divisibility easy to inspect. A candidate divisor can use only primes already in the factorization and cannot exceed any listed exponent. For 360, 2³ × 3 is allowed, but 2⁴ is not. Prime powers also provide the building blocks for finding greatest common divisors and least common multiples.
Why the input range is bounded
Trial division is predictable for the accepted range but is not an appropriate method for cryptographic-size integers. The maximum is one trillion, below JavaScript’s exact-integer limit. This tool rejects decimals, negative values, zero, and one rather than describing them as composite. One is neither prime nor composite.
Assumptions & limitations
What this calculation assumes
- Input is an integer between 2 and one trillion.
What to keep in mind
- The tool counts divisors but does not enumerate every divisor or factor arbitrarily large integers.
Common questions
Can a prime number have a prime factorization?
Yes. Its factorization is the number itself, with exponent 1. It has two positive divisors: 1 and itself.
Does the order of prime factors matter?
No. Multiplication is commutative. The table lists factors in increasing order to make repeated primes easier to compare.