Prime Factorization Calculator writes a nonzero integer as prime powers and repeated factors, then checks the complete positive divisor set.
Read prime powers and repeated factors
Prime factorization writes a nonzero integer’s magnitude as a product of primes. OpenStax’s prime-factor chapter explains the uniqueness of that product for integers greater than one, apart from the order of the primes. Exponents record how many times each prime appears. Enter one integer. The primary answer shows prime powers; calculation details show the repeated factors, division steps and complete positive divisor list. The tool tests divisibility by the smallest primes, divides out every occurrence, then continues up to the square root of the remaining value. Any final remaining value is prime. The result is checked by multiplying its prime powers back together.
Keep the original signs and omit thousands separators, decimal points and scientific notation.
Review the named primary result and open calculation details to check the arithmetic.
Prime factorization of 360: worked example
The preset enters 360 and returns 2^3 × 3^2 × 5. Its repeated-factor form is 2 × 2 × 2 × 3 × 3 × 5. There are 3 distinct primes and 6 prime factors counting repeats.
360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5
The complete positive divisor count is 24; their sum is 1170. These counts describe different objects from the prime-only list.
Prime factors compared with the full divisor list
A prime factor is prime; a divisor can be composite or one. Every positive divisor is obtained by choosing an exponent from zero through the corresponding exponent in the prime factorization. Choosing all possible combinations produces the complete positive divisor set. MathWorld’s Divisor Function reference gives the count as the product of one plus each prime exponent.
For a prime input, the prime preset returns 97. Its positive divisor list is 1, 97, and the divisor count is 2. The detailed divisor output checks the decomposition. A full divisor list answers a different question from a prime-only factorization; factor pairs describe products rather than prime multiplicities.
One, zero, negative integers and the input range
One is neither prime nor composite. Entering 1 produces 0 prime factors and the positive divisor list 1. This is the empty-prime-product case, not a claim that one is prime. Zero is rejected because it has no finite complete prime decomposition or divisor list.
For negative inputs, the sign is factored out separately. The signed preset gives -84 = -1 × 2^2 × 3 × 7. The primes decompose the positive magnitude; the negative sign is not an additional prime.
This tool accepts nonzero integers with at most 14 digits, using exact integer input and a complete bounded decomposition. Decimal points, fractions, exponents, unsafe numeric API values and malformed text are rejected. The range is a local computation policy, not a claim that larger integers cannot be factored. Division steps provide a valid factorization path; a branching graphical factor tree is not part of this interface.
Greatest Common Factor Calculator finds the greatest divisor shared by an integer list; Least Common Multiple Calculator finds the smallest positive shared multiple of nonzero integers, with an explicit zero convention.
A complete divisor list and product-checked factor pairs belong in the Factors Calculator, which also explains signed inputs.
When you need coefficients that reproduce the GCD, the Extended Euclidean Algorithm Calculator checks the original-input identity and its inverse residue.