Prime Factorization Calculator

Prime factorization expresses a positive integer as a product of prime numbers. A prime number has exactly two positive divisors, one and itself, so primes act as the indivisible building blocks of whole-number multiplication. For example, 360 can be written as 2 × 2 × 2 × 3 × 3 × 5, or more compactly as 2³ × 3² × 5. The fundamental theorem of arithmetic guarantees that every integer greater than one has exactly one prime factorization apart from the order of its factors. The calculator uses trial division. It first removes every factor of two, recording how many times the division succeeds. It then tests odd candidate divisors in increasing order. Testing can stop when the candidate squared exceeds the remaining number, because any composite remainder would have had a factor no larger than its square root. If a remainder greater than one survives, that remainder is prime and becomes the final factor. Exponents make repeated factors easy to read. In 2³, the exponent three means that three copies of two are multiplied. A prime that occurs once is displayed without an exponent. The reported distinct-factor count tells you how many different primes occur, not how many prime factors occur when repetitions are counted. Thus 360 has three distinct prime factors but six factors when multiplicity is included. Prime factorization is useful for reducing fractions, finding greatest common divisors and least common multiples, identifying perfect squares or cubes, and solving divisibility problems. To find a greatest common divisor, compare factorizations and take each shared prime at its smaller exponent. To find a least common multiple, take every prime that appears at its larger exponent. A number is a perfect square exactly when all exponents in its prime factorization are even. Cryptography also relies on the difference between multiplication and factorization. Multiplying two large primes is straightforward, while recovering them from their product can be computationally difficult. This browser calculator is designed for ordinary educational and practical integers, not for attacking cryptographic key sizes. Very large prime inputs can require many trial divisions, although obvious small factors are removed quickly. The domain begins at two because one is neither prime nor composite and has an empty prime product. Zero and negative integers need additional conventions: zero has no prime factorization, while a negative integer can be represented by -1 times the factorization of its absolute value. Enter a positive integer here to receive the standard unique form used in arithmetic and number theory.