What this calculator does
This tool finds every factor (divisor) of a whole number and lists all of its factor pairs — the pairs of numbers that multiply together to give your number. A factor of an integer is a whole number that divides it exactly, leaving no remainder. Factor pairs are simply the two factors that multiply to produce the original value, such as \(6 \times 8 = 48\).
How to use it
Type any nonzero integer (positive or negative) into the box and submit. The calculator returns three things: how many factors the number has, the complete list of factors in ascending order, and every factor pair written as \(a \times b = n\). For negative numbers it shows signed pairs, because a negative product needs one negative and one positive factor.
The formula explained
The calculator uses trial division. Let \(n\) be your number and \(m = |n|\) its absolute value. It only needs to test candidate divisors \(i\) from 1 up to the floor of the square root of \(m\). Whenever \(m \bmod i\) equals 0, both \(i\) and \(\tfrac{m}{i}\) are factors, and together they form the pair $$i \times \tfrac{m}{i} = m.$$ Testing only up to \(\sqrt{m}\) makes the search fast, because every factor above the square root is automatically paired with one below it. Perfect squares list their square root once but show it paired with itself.
Worked example
For \(n = 48\), the square root is about 6.93, so we test \(i = 1\) to 6. We find $$1\times48,\; 2\times24,\; 3\times16,\; 4\times12 \text{ and } 6\times8$$ (5 does not divide 48). The factor list is 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — ten factors in total.
FAQ
Why can't I enter 0? Every integer divides 0, so 0 would have infinitely many factors. Enter a nonzero integer instead.
How do negative numbers work? For -6 the product must be negative, so each positive pair becomes two signed pairs: \(-1\times6,\; 1\times-6,\; -2\times3,\; 2\times-3\).
What is a prime number here? A prime has exactly two factors: 1 and itself, for example 7 gives 1 and 7.
