Find prime factors with exponents or as an expanded product. Follow division steps, check the product and list positive divisors for integers from 2 to 10¹².
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| n | p | n ÷ p |
|---|---|---|
| 360 | 2 | 180 |
| 180 | 2 | 90 |
| 90 | 2 | 45 |
| 45 | 3 | 15 |
| 15 | 3 | 5 |
| 5 | 5 | 1 |
1 · 2 · 3 · 4 · 5 · 6 · 8 · 9 · 10 · 12 · 15 · 18 · 20 · 24 · 30 · 36 · 40 · 45 · 60 · 72 · 90 · 120 · 180 · 360
Enter an integer from 2 to 1,000,000,000,000. The calculator divides by 2 and then tests odd divisors up to the square root of the remaining value. It shows the prime factors with exponents, the expanded product, division steps and positive divisors. Multiply the prime factors to check the original number; 1 is a divisor but is not a prime factor.
Recomputed from this page's implementation at build time · checked 2026-10-22
Divisor counts come from exponents, not from the size of the number. That is why 720,720 beats 1,000,000,000,000 on divisor count while being less than a millionth of its size — and why the slowest input this page accepts is the one that refuses to break apart at all.
| 360 | 2,520 | 5,040 | 720,720 | 10^9 | |
|---|---|---|---|---|---|
| Number of divisors | 24 | 48 | 60 | 240 | 100 |
720,720 has 240 divisors while a billion has 100, even though the billion is a thousand times larger. The count follows how the primes are spread - 720,720 carries six distinct ones - not the size of the number.
Six prime factors but only three distinct primes, and 24 divisors is 4 × 3 × 2: add one to each exponent and multiply. That identity is the reason a divisor count can be produced without ever listing the divisors — and it is the fastest way to sanity-check any factorisation you get.
Only two distinct primes, yet 169 divisors, because 13 × 13 multiplies two large exponents. Note the last row: the divisors add up to more than twice the number itself, since the number counts as one of its own divisors.
Set this next to the previous case. 720,720 is under a millionth the size and still carries more divisors, because its six primes are the smallest ones available. Divisor count rewards many small primes rather than a big number, which is why highly composite numbers always factor into 2, 3, 5, 7, 11 and 13 in some arrangement.
The divisor sum is exactly twice the number, which is the definition of a perfect number. 8,128 is 2^6 × (2^7 − 1), and the shape generalises: an even perfect number is always a power of two multiplied by a Mersenne prime. If a divisor sum ever looks implausible, this is the cheapest test available — for a perfect number it must equal 2n.
Allow roughly 28 ms for this one against well under a millisecond for the trillion above, because trial division only stops when the divisor exceeds the square root of whatever is left to factor. When the second-largest factor is the number itself, nothing shortens the loop. The answer is still trivial to read: a twelve-digit prime, so the only divisors are 1 and itself.
inputError
One
1 is the empty product — it has no prime factorisation at all. Returning 1 = 1 would be technically true and completely useless, so the input is refused instead.
inputError
One and a half
Whole numbers only. Rounding 1.5 up to 2 would answer a question that was never asked, so a fractional input is rejected rather than coerced into the nearest integer.
inputError
A trillion and one
The cap is a time limit rather than a display limit. Trial division runs to the square root of the remaining value, so ten times the input costs about three times the wait; 10¹² is where the worst case stays inside a fraction of a second.
inputError
Minus six
Primality is defined on positive integers. The sign is not stripped anywhere, so a negative entry is refused rather than silently factored as its absolute value.
Calculator Soup prime factorschecked 2026-10-22
Why they built it that wayThe output leads with a long-division ladder rather than the exponent form. That suits a teaching page where the steps are the answer, and it is why large inputs are slow: every division gets printed.
Divisor count and divisor sum cannot be read off a factor tree — they follow from the exponents, and computing them is what makes the perfect-number check possible. Their cap is ten times higher; ours is set where trial division still finishes quickly, and the trade is stated rather than hidden.
Enter 720720 and read the exponents: 4, 2, 1, 1, 1, 1. Add one to each and multiply — 5 × 3 × 2 × 2 × 2 × 2 = 240, which is the divisor count shown. Then enter 999999999989: the factorisation returns the number unchanged, the divisor count drops to 2, and the divisor sum becomes the number plus one.
Multiplied every factorisation back to its input and checked divisor counts with the exponent formula · impl 7934c399
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
n = Π p_i^a_i; τ(n) = Π(a_i + 1); σ(n) = Π Σ_{j=0}^{a_i} p_i^j
Uses trial division for integers from 2 to 1,000,000,000,000, testing 2 then odd divisors up to the square root of the remaining value. Larger factors may take longer to find.
Enter an integer from 2 to 10¹². For example, 360 = 2³ × 3² × 5. Follow the division steps and multiply the factors to check the original number. The positive-divisor list includes composite divisors as well as primes.
Positive divisors divide the original number without a remainder. They may be prime or composite and include 1 and the number itself. For 360 = 2³ × 3² × 5, the number of positive divisors is (3 + 1) × (2 + 1) × (1 + 1) = 24.