How it works
Every positive integer has a set of whole numbers that divide it with no remainder. This calculator takes a single input and returns that complete, sorted list—useful for simplifying fractions, checking divisibility, or solving number-theory problems.
The efficient approach avoids testing every number from 1 upward. Instead, the tool checks potential divisors only up to the square root of the input. Each time it finds a divisor, it also records the matching quotient, capturing factors in pairs. This cuts the work dramatically: for 360, only 19 trial divisions are needed instead of 360.
The formula
Factors of n = { d : d ∈ ℤ⁺, d ≤ √n, n mod d = 0 } ∪ { n / d : d ∈ ℤ⁺, d ≤ √n, n mod d = 0 }
In plain terms: test each whole number d from 1 up to √n. Whenever n ÷ d leaves no remainder, both d and n ÷ d are factors. Collect every such pair, sort them, and you have the full divisor list.
Worked example
Finding all factors of 360.
First, compute the square root: √360 ≈ 18.97. So trial divisors run from 1 through 18.
| Trial divisor d | 360 ÷ d | Remainder | Factor pair found |
|---|---|---|---|
| 1 | 360 | 0 | 1, 360 |
| 2 | 180 | 0 | 2, 180 |
| 3 | 120 | 0 | 3, 120 |
| 4 | 90 | 0 | 4, 90 |
| 5 | 72 | 0 | 5, 72 |
| 6 | 60 | 0 | 6, 60 |
| 8 | 45 | 0 | 8, 45 |
| 9 | 40 | 0 | 9, 40 |
| 10 | 36 | 0 | 10, 36 |
| 12 | 30 | 0 | 12, 30 |
| 15 | 24 | 0 | 15, 24 |
| 18 | 20 | 0 | 18, 20 |
Divisors 7, 11, 13, 14, 16, and 17 each leave a remainder, so they are skipped. When d and the quotient coincide (as with perfect squares), the pair collapses to a single factor—360 is not a perfect square, so every pair here has two distinct members.
Collecting and sorting every value from the right-hand column gives the complete factor list:
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
That is 24 divisors in total.
Things to watch
Duplicate pairs at perfect squares. If your input is a perfect square (say, 144), the divisor 12 pairs with itself (144 ÷ 12 = 12). A correct factor list shows 12 only once. The calculator handles this automatically, but it is a common source of error in manual work.
Including 1 and the number itself. Some problems ask for proper factors, which exclude the number itself; others ask for non-trivial factors, which exclude both 1 and n. The full list this tool returns always includes both endpoints—filter as needed.
Negative divisors. Every positive factor has a negative counterpart (−2 also divides 360). Most practical applications—simplifying fractions, arranging items in grids—use positive divisors only, so this calculator restricts output to positive integers.
Prime factorization is a different operation. Breaking 360 into 2³ × 3² × 5 reveals the building blocks, but it does not directly enumerate composite divisors like 24 or 45. You still need the pairing method above to surface every divisor.
This calculator gives exact arithmetic results for integer inputs. For very large numbers (billions and beyond), computation time may increase, but the underlying math remains exact—no rounding or approximation is involved.