Factors calculator guide
A factor — also called a divisor — of a positive integer n is a positive integer d such that n = d × k for some integer k. In other words, d divides n with no remainder. Every whole number has 1 and itself as factors. Numbers with exactly two factors are prime; numbers with more than two factors are composite; 1 has only one factor and is neither prime nor composite.
This calculator returns the complete list of positive factors, not the prime factorization. The factors of 12 are 1, 2, 3, 4, 6, and 12. The prime factors of 12 are only 2 and 3, because 12 = 2² × 3. Use this page when you need every divisor; use a prime-factorization or GCF/LCM tool when you need the prime breakdown or a shared factor of several numbers.
How to use this factors calculator
- Enter a positive integer: Type a whole number of 1 or greater, for example 120. Zero, negatives, decimals such as 12.5, and empty input are rejected.
- Select Calculate: Press Calculate to list every factor. The default example 120 is already evaluated when the page loads.
- Read the count, the list, and the prime check: The large number is how many factors were found. Below it, the factors appear in ascending order, separated by commas. A short line states whether the input is prime (exactly two factors).
- Reset or share: Use Reset to return to 120, or copy the link to share the current number — the input is stored in the page URL.
Formula and variables
Trial division finds every factor by testing integers from 1 up to √n. Whenever i divides n, both i and n / i are factors, so the search never needs to go past the square root. If n is a perfect square, √n is a factor and is counted only once. The count of factors also follows from the prime factorization: raise each prime exponent by one and multiply. That count is a check on the list, not a substitute for listing the divisors themselves.
d | n ⇔ n = d × k for an integer k; if n = p₁^a × p₂^b × … then the number of factors is (a+1)(b+1)…- n — Input integer
- The positive whole number whose factors are listed. (unitless integer)
- d — Factor (divisor)
- A positive integer that divides n with no remainder. (unitless integer)
- k — Cofactor
- The complementary factor so that d × k = n. (unitless integer)
- pᵢ^a — Prime-power in the factorization
- Used only to count factors: if n = 2³ × 3 × 5, there are (3+1)(1+1)(1+1) = 16 factors. (unitless integer)
Worked example: all factors of 120
List every positive factor of the default example 120 using trial division up to √120, which is the same method the calculator applies.
- Positive integer
- 120
- √120 ≈ 10.95, so test whole numbers from 1 through 10.
- 120 ÷ 1 = 120, so 1 and 120 are factors.
- 120 ÷ 2 = 60, so 2 and 60 are factors.
- 120 ÷ 3 = 40, so 3 and 40 are factors.
- 120 ÷ 4 = 30, so 4 and 30 are factors.
- 120 ÷ 5 = 24, so 5 and 24 are factors.
- 120 ÷ 6 = 20, so 6 and 20 are factors.
- 120 ÷ 7 and 120 ÷ 9 leave remainders; skip them. 120 ÷ 8 = 15 and 120 ÷ 10 = 12, adding 8, 15, 10, and 12.
- Prime-factorization check: 120 = 2³ × 3 × 5, so the number of factors is (3+1)(1+1)(1+1) = 16. Those 16 numbers are all factors of 120; only 2, 3, and 5 are prime factors.
Result: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 — 16 factors; 120 is not prime
Sixteen positive integers divide 120 evenly. Because there are more than two, 120 is composite. Do not confuse this list with the prime factorization 2³ × 3 × 5: 4, 6, 8, 10, and the rest are factors, but they are not prime.
Understanding your results
Factor list
The comma-separated values are every positive divisor of n, written from smallest to largest. The first entry is always 1 and the last is always n. Factor pairs that multiply to n sit at opposite ends of the list: for 12, 1 pairs with 12, 2 with 6, and 3 with 4.
Factor count and primality
A count of 2 means n is prime: its only factors are 1 and itself. A count greater than 2 means n is composite. A count of 1 occurs only for n = 1, which is neither prime nor composite. Perfect squares have an odd count because one factor (the square root) pairs with itself; 36 has nine factors, including 6.
Factors are not the same as prime factors
Prime factors are the prime numbers that multiply to n. Every prime factor is a factor, but most factors are products of those primes. For 12 the factors are 1, 2, 3, 4, 6, 12; the prime factors are 2 and 3. This calculator reports the full list. Use a prime-factorization tool if you need 12 = 2² × 3.
Assumptions
- The input is a positive whole number (an integer of 1 or greater).
- Only positive factors are listed; negative divisors such as −3 for 12 are omitted, matching elementary usage.
- n is prime exactly when it has two factors. 1 is reported as not prime.
- Inputs larger than 10,000,000 are rejected so trial division finishes quickly.
Limitations
- The calculator lists positive factors only; it does not print the prime factorization, a factor tree, or the GCF of several numbers.
- Zero, negatives, and non-integers are outside the domain and produce an error instead of a truncated result.
- Values above 10,000,000 are rejected rather than factored with a slower algorithm.
- Polynomial factors, rational factors, and factorization over the integers of several variables are not supported.
Common mistakes
- Confusing the full factor list with the prime factorization: 4 is a factor of 12, but it is not a prime factor.
- Stopping the search too early and missing the cofactor of a small divisor (if 3 divides 24, then 8 does too).
- Testing past the square root and double-counting pairs that were already found.
- Calling 1 a prime number because it “cannot be divided further”; a prime must have exactly two distinct factors.
- Treating 0 as having factors; division by zero is undefined, and 0 is not a positive integer here.
- Expecting decimals such as 2.5 to have a whole-number factor list; factors of integers are themselves integers.
Practical use cases
Check whether a number is prime
If the calculator reports exactly two factors, the number is prime. If it reports more than two, the number is composite and the extra factors are shown so you can see why.
Simplify a fraction
The factors of the numerator and denominator include every candidate for cancellation. For 12/30 the shared factors include 2, 3, and 6; dividing both parts by 6 reduces the fraction to 2/5.
Arrange items into equal groups or rectangles
Each factor pair of 24 is a rectangular layout: 1×24, 2×12, 3×8, and 4×6. The same list tells how many equal teams 24 people can form.
Prepare a GCF or LCM calculation
Listing the factors of each number is the set-intersection method for the GCF. For two or more numbers at once, the dedicated GCF and LCM calculator is faster; this page is the right tool for one integer.
Planning and decision guide
Stop at the square root
Once i exceeds √n, every remaining factor is already the cofactor of a smaller i. For 100 you only need to test through 10. The calculator uses this bound automatically; use it by hand to keep homework short.
Use divisibility rules before dividing
Even numbers are divisible by 2. A number is divisible by 3 if the sum of its digits is. It is divisible by 5 if it ends in 0 or 5. Applying those checks first skips many trial divisions.
Choose the right related tool
This calculator answers “what divides n?” For “what is n as a product of primes?” use prime factorization. For “what is the largest number that divides several integers?” use the GCF and LCM calculator.
Frequently asked questions
What is a factor of a number?
A factor (divisor) of a positive integer n is a positive integer that divides n with no remainder. For 12 the factors are 1, 2, 3, 4, 6, and 12, because each of those products is 12: 1 × 12, 2 × 6, and 3 × 4.
What is the difference between factors and prime factors?
Factors are every positive divisor of n. Prime factors are only the prime numbers in that list — the primes that multiply to n. For 12 the factors are 1, 2, 3, 4, 6, 12; the prime factors are 2 and 3, since 12 = 2² × 3. 1 is a factor of every number but is never a prime factor.
How does the calculator find all factors?
It uses trial division up to the square root of n. For each integer i from 1 through √n that divides n evenly, both i and n / i are recorded. The list is then sorted from smallest to largest. You only need to search that far because larger factors are cofactors of smaller ones already found.
How many factors does a number have?
It depends on n. Primes have exactly two factors. Perfect squares have an odd count because the square root pairs with itself. If the prime factorization is n = p₁^a × p₂^b × …, the number of positive factors is (a+1)(b+1)…. For 120 = 2³ × 3 × 5 that product is 4 × 2 × 2 = 16.
Is 1 a prime number?
No. A prime number must have exactly two distinct positive factors: 1 and itself. 1 has only one factor, so it is neither prime nor composite. Treating 1 as prime would also break unique prime factorization, because you could insert extra factors of 1 indefinitely.
What is the difference between factors and multiples?
Factors divide into a number; multiples are produced by multiplying it. Factors of 12 are 1, 2, 3, 4, 6, 12 (all ≤ 12). Multiples of 12 are 12, 24, 36, 48, … (all ≥ 12, and there are infinitely many).
Can negative numbers or zero be factors here?
Mathematically −3 divides 12, but this calculator lists only positive factors, which is the usual convention in elementary number theory and schoolwork. Zero is not accepted as an input: a positive integer of 1 or greater is required, and division by zero is undefined.
Why is there a 10,000,000 limit?
The search tests every integer up to √n. For n = 10,000,000 that is about 3,162 divisions, which stays instant in the browser. Larger n is rejected rather than making the page wait on a longer loop.
Sources and review
- Divisor — MathWorld — A Wolfram Resource. Accessed 2026-08-23.
- Factors of a Number — All Factors — Math Is Fun. Accessed 2026-08-23.
- Factors and multiples — Khan Academy. Accessed 2026-08-23.
Reviewed 2026-08-23.