Factor Calculator
Every factor of a number, its prime factorisation, factor pairs, the divisor count and sum, and whether it is prime, perfect, abundant or deficient.
Factor Calculator: with the default inputs, all factors is 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
Any whole number from 1 up to a trillion.
- Number of factors
- 24Written d(n) or τ(n).
- Sum of all factors
- 1,170Written σ(n); includes the number itself.
- Prime factorisation
- 2³ × 3² × 5
- Distinct prime factors
- 3
- Aliquot sum (proper factors)
- 810Everything except the number itself.
- Classification
- Composite and abundant — its proper factors add up to more than itself
- Largest proper factor
- 180
Assumptions
- Positive whole numbers only, up to 10¹².
- Trial division to √n, which is exhaustive — primality here is proved, not estimated.
- 'Factors' means positive divisors, including 1 and the number itself.
| Factor | × partner | Product | |
|---|---|---|---|
| 1 | 360 | 360 | |
| 2 | 180 | 360 | |
| 3 | 120 | 360 | |
| 4 | 90 | 360 | |
| 5 | 72 | 360 | |
| 6 | 60 | 360 | |
| 8 | 45 | 360 | |
| 9 | 40 | 360 | |
| 10 | 36 | 360 | |
| 12 | 30 | 360 | |
| 15 | 24 | 360 | |
| 18 | 20 | 360 |
12 pairs. Factors come in pairs around √n = 18.9737, which is why you only ever have to test up to the square root.
| Prime | Exponent | Prime power | Choices (exponent + 1) |
|---|---|---|---|
| 2 | 3 | 8 | 4 |
| 3 | 2 | 9 | 3 |
| 5 | 1 | 5 | 2 |
Multiply the "choices" column: 4 × 3 × 2 = 24 divisors. Each divisor picks one exponent from 0 up to e for every prime, independently.
How this is worked out
The formula
If n = p₁^e₁ × p₂^e₂ × … × p_k^e_k (the unique prime factorisation), then number of divisors d(n) = (e₁ + 1)(e₂ + 1) … (e_k + 1) sum of divisors σ(n) = ∏ (pᵢ^(eᵢ+1) − 1) ÷ (pᵢ − 1) aliquot sum s(n) = σ(n) − n s(n) = n → perfect, s(n) > n → abundant, s(n) < n → deficient Factors pair up: if d divides n then so does n ÷ d, and one of the pair is ≤ √n. That is why testing divisors only up to √n finds all of them.
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Number to factor
- Any whole number from 1 up to a trillion.from 1 to 1000000000000 · whole numbers only · defaults to 360
What you get back
- All factorsmain answer
- Number of factors
- Written d(n) or τ(n).
- Sum of all factors
- Written σ(n); includes the number itself.
- Prime factorisation
- Distinct prime factors
- Aliquot sum (proper factors)
- Everything except the number itself.
- Classification
- Largest proper factor
What this assumes
- Positive whole numbers only, up to 10¹².
- Trial division to √n, which is exhaustive — primality here is proved, not estimated.
- 'Factors' means positive divisors, including 1 and the number itself.
About this calculator
Give it a number and it returns everything the divisibility questions ask for: the complete list of factors, the factor pairs, the prime factorisation, how many divisors there are, what they add up to, and which of the classical categories the number falls into.
Factors and factor pairs
A factor (or divisor) of n is a whole number that divides n with nothing left over. They always come in pairs — if 8 divides 360 then so does 45, because 8 × 45 = 360 — and in every pair one member is at or below √n. That single observation is what makes factoring fast: to find every factor of 360 you only have to test up to 18.97, not up to 360. The pair table lays this out; a perfect square is the one case where a factor pairs with itself.
Prime factorisation
Every whole number above 1 breaks into primes in exactly one way, a fact solid enough to be called the fundamental theorem of arithmetic. 360 is 2³ × 3² × 5, and no other multiset of primes multiplies to 360. This is why 1 is not counted as prime: if it were, you could pad any factorisation with as many 1s as you liked and uniqueness would collapse.
The factorisation also predicts the factor count without listing anything. A divisor of 2³ × 3² × 5 is built by choosing an exponent from 0 to 3 for the 2, from 0 to 2 for the 3, and from 0 to 1 for the 5 — that is 4 × 3 × 2 = 24 combinations, so 360 has 24 factors. The same trick, applied to geometric series, gives the sum of the divisors in closed form.
Perfect, abundant and deficient
Add up every factor except the number itself — the aliquot sum. If it lands exactly on the number, the number is perfect: 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14. Perfect numbers are extraordinarily rare; only 51 are known, all even, and whether an odd one exists is a question that has been open for over two thousand years. If the aliquot sum overshoots, the number is abundant (12, 18, 20 and every multiple of a perfect number); if it falls short, deficient (every prime, every prime power, and most numbers you will ever type).
Limits and honesty about the algorithm
Trial division up to the square root is used, which is exhaustive: when the calculator says a number is prime, that is a proof, not a probabilistic test. The trade-off is speed, so the input is capped at a trillion — √10¹² is a million, which is instant. Beyond that, factoring stops being trivial, and the difficulty of factoring genuinely large numbers is what most public-key cryptography rests on.
Frequently asked questions
▸How many factors does 60 have?
Twelve: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. They sum to 168. The prime factorisation 2² × 3 × 5 predicts the count directly as 3 × 2 × 2 = 12.
▸Is 1 a prime number?
No. A prime has exactly two distinct positive factors; 1 has only one. Excluding it is what makes prime factorisation unique.
▸What is a perfect number?
One whose proper factors add up to itself — 6, 28, 496, 8128, and then a very long gap. Only 51 are known, all even, and nobody has ever found an odd one or proved that none exists.
▸Why only test divisors up to the square root?
Because factors pair up around √n. If n had a factor larger than √n with no partner at or below √n, their product would exceed n. So every factor is found by scanning to the square root and reading off both members of each pair.
▸Does this handle negative numbers?
Enter the absolute value. Every factor d of n is matched by −d, so the factor list of −360 is the list shown here with a negative copy of each entry.
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