Factoring Calculator
Find every factor of a number — all divisors and factor pairs — with step-by-step trial division up to the square root. Negatives and primes handled.
How to use this calculator
Type any whole number — positive or negative — into the box. The calculator lists every factor (divisor) in ascending order, groups them into the factor pairs that multiply back to your number, and shows the trial-division working underneath. Prime numbers are flagged, and negative inputs also get their sign-flipped pairs.
What is a factor?
A factor (or divisor) of a whole number is an integer that divides evenly — with no remainder. Because , both 6 and 8 are factors of 48. Every number has at least two factors, 1 and itself; a number whose only factors are 1 and itself is prime.
Why trial division stops at the square root
Factors always appear in pairs . If both members of a pair were bigger than , their product would be bigger than — impossible. So at least one factor of every pair is less than or equal to .
That is why the calculator only tests . Each divisor it finds is paired with its quotient , which recovers the larger factor for free. For 48, , so testing 1 through 6 is enough to discover all ten factors.
Factoring negative numbers
The divisors of a negative number are the same as those of its absolute value. What changes is the sign of the pairs: since a negative times a positive is negative, each pair has a sign-flipped twin. For :
Factors vs. multiples
These are easy to mix up:
| Factors of 12 | Multiples of 12 | |
|---|---|---|
| Definition | divide into 12 | 12 times an integer |
| Values | 1, 2, 3, 4, 6, 12 | 12, 24, 36, 48, … |
| Size | never larger than 12 | never smaller than 12 |
| How many | finite | infinite |
Worked example — the factors of 48
- , so test the integers 1 through 6.
- → pair (1, 48).
- → pair (2, 24).
- → pair (3, 16).
- → pair (4, 12).
- 5 does not divide 48 evenly — skip it.
- → pair (6, 8).
Collecting both members of every pair gives the ten factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Frequently asked questions
- What are the factors of 48?
- 48 has ten factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. They come in pairs that multiply to 48: (1, 48), (2, 24), (3, 16), (4, 12), and (6, 8).
- What is a factor of a number?
- A factor (or divisor) of a whole number is an integer that divides with no remainder. For example 6 is a factor of 48 because exactly, while 5 is not because leaves a remainder.
- Why does the calculator only test numbers up to the square root?
- Factors always come in pairs . If both were larger than their product would exceed , so one factor of every pair is at or below . Testing up to and pairing each hit with its quotient finds them all.
- Can a negative number be factored?
- Yes. The divisors of a negative number are the same as those of its absolute value, but each factor pair can also be written with opposite signs, since a negative times a positive is negative. For −6 the pairs are (1, −6), (−1, 6), (2, −3), and (−2, 3).
- Why is 0 not allowed?
- Every integer divides 0 (because for any ), so 0 has infinitely many factors and no finite list to show. The calculator asks for a non-zero number instead.
- What is the difference between a factor and a multiple?
- A factor divides into the number (factors of 12 are 1, 2, 3, 4, 6, 12). A multiple is the number times an integer (multiples of 12 are 12, 24, 36, …). Factors are never larger than the number; multiples are never smaller.
- How can I tell if a number is prime from its factors?
- A prime number has exactly two factors: 1 and itself. If the factor list has only two entries, the number is prime. 17 → {1, 17} is prime; 18 → {1, 2, 3, 6, 9, 18} is not.