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Prime Factorization Calculator

Enter a number to see its prime factorization

Result

2³ × 3² × 5
2 × 2 × 2 × 3 × 3 × 5
3

How it is calculated

The tool uses trial division. It tries each divisor d from 2 upward, divides the number by d as many times as it goes in evenly and counts how many times, then moves on. It stops when d × d is larger than what is left; any remainder above 1 is itself a prime. The result is shown as a product with exponents, as the list of every factor written out, and as the count of distinct primes.

The input must be a whole number of 2 or more. Decimals such as 1.5, negative numbers, 0 and 1 give no result because they have no prime factorization. A value like 360.0 is accepted as 360. Very large numbers become slow, and from 2⁵³ (9,007,199,254,740,992) upward the tool gives no result, because a number that large cannot be held exactly.

Worked example

Factor 360. Divide by 2: 180, 90, 45, so 2 goes in three times and leaves 45. Divide by 3: 15, then 5, so 3 goes in twice and leaves 5. Since 5 is larger than the remaining search limit, 5 is prime itself. The compact form is 2³ × 3² × 5, with 3 distinct primes, and written out in full it is 2 × 2 × 2 × 3 × 3 × 5. Check: 8 × 9 × 5 = 360.

Try this example →

Common mistakes

  • Entering 1 and expecting a factorization. 1 is not a prime and has no prime factors, so the tool shows nothing.
  • Counting the factor list instead of the distinct primes. The count shows how many different primes appear (3 for 360), not the number of factors written out (6).
  • Mixing up prime factors with all divisors. 360 has many divisors, such as 4, 6 and 12, but only 2, 3 and 5 are prime.
  • Entering 9,007,199,254,740,992 (2⁵³) or more and expecting a factorization. The tool stops there and shows no result, because such a number cannot be held exactly.
  • Typing a number with a thousands separator or a decimal part and expecting it to be read; only plain whole numbers work.

Sources

About this calculator

This calculator breaks any whole number down into its prime factors using trial division — testing 2, 3, 5, 7 and onward until only primes remain — and shows both the expanded product and the compact exponent form (like 2³ × 3² × 5).

Math students use it to check factoring homework, and prime factorization is the foundation of topics like finding the GCD/LCM of numbers and the basic number theory that underlies public-key cryptography like RSA.

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Comments (1)

  • Editorial team

    Factorizations give you the greatest common divisor and least common multiple directly. Take 360 = 2³ × 3² × 5 and 84 = 2² × 3 × 7: the GCD uses the smallest exponent of shared primes, 2² × 3 = 12, and the LCM uses the largest of every prime, 2520. Inputs of 2⁵³ (about nine quadrillion) or more give no result rather than a rounded one. What do you need the factors for?

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