Factor Calculator

Enter a positive whole number to find all of its factors (divisors), its prime factorization, factor count, and factor sum.

Quick Facts

Definition
a is a factor of n if n ÷ a leaves no remainder
Factors always come in pairs that multiply to n.
Search range
Check i = 1 to √n
Every factor pairs with n ÷ i, so testing past √n is unnecessary.
Prime factorization
n = p₁^e₁ × p₂^e₂ × …
Unique for every integer > 1 (Fundamental Theorem of Arithmetic).

Your Results

Calculated
All Factors
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Every whole-number divisor
Prime Factorization
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Product of prime building blocks
Number of Factors
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Count of divisors
Sum of Factors
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Total of all divisors

Ready

Enter a whole number and press Calculate.

How the Factor Calculator works

A factor (or divisor) of a whole number n is any whole number a such that n ÷ a comes out even, with no remainder — equivalently, n = a × b for some whole number b. This calculator takes a positive integer and finds every one of its factors, breaks it down into its prime factorization, and reports how many factors it has and what they add up to.

The trial-division method, step by step

  • Test each whole number i starting at 1 and going up to √n.
  • If n ÷ i has no remainder, then i is a factor — and so is its pair, n ÷ i, because i × (n ÷ i) = n.
  • Collect every pair found, remove any duplicate when n is a perfect square (√n pairs with itself), and sort the results.
  • Only checking up to √n is enough: if a factor were larger than √n, its paired factor would have to be smaller than √n, so it would already have been found.

Prime factorization

  • Building blocks: the prime factorization writes n as a product of primes, e.g. 60 = 2² × 3 × 5. Every integer greater than 1 has exactly one such factorization (the Fundamental Theorem of Arithmetic), regardless of the order primes are found in.
  • Finding it: divide n by the smallest possible prime (2, then 3, then 5, …) repeatedly, moving to the next prime whenever the current one no longer divides evenly, until what remains is 1.
  • Counting factors from primes: if n = p₁^e₁ × p₂^e₂ × …, the total number of factors equals (e₁+1) × (e₂+1) × … — for 60 = 2² × 3¹ × 5¹, that's 3 × 2 × 2 = 12 factors.

Common uses

Factoring is the basis for simplifying fractions, finding the greatest common factor (GCF) or least common multiple (LCM) of two numbers, reducing radicals, and factoring polynomials in algebra. It also shows up directly in cryptography, scheduling problems (finding numbers that divide evenly into a time period), and puzzle or number-theory work where knowing exactly how a number can be split apart matters.

Frequently Asked Questions

What is a factor of a number?
A factor (or divisor) of a whole number n is any whole number that divides n exactly, leaving no remainder. For example, 6 is a factor of 24 because 24 ÷ 6 = 4 with no remainder. Every positive integer has at least two factors, 1 and itself.
How do you find all the factors of a number?
Test every whole number i from 1 up to √n. Whenever n ÷ i has no remainder, both i and n ÷ i are factors. Checking only up to the square root works because factors always come in pairs that multiply to n, so you find every pair without testing all the way up to n.
What is prime factorization and how is it different from a list of factors?
Prime factorization writes a number as a product of prime numbers, such as 24 = 2³ × 3. The full list of factors (1, 2, 3, 4, 6, 8, 12, 24) includes every divisor, while the prime factorization shows only the prime "building blocks" that generate them. Every integer greater than 1 has exactly one prime factorization — the Fundamental Theorem of Arithmetic.
Can a number have negative factors, and what about 0 or 1?
Yes — for every positive factor f of n, −f is also a factor, since n ÷ (−f) is also a whole number. This calculator lists positive factors by default and can include the matching negative factors. The number 1 has exactly one factor (itself); 0 is divisible by every nonzero number, so it is not given a standard finite factor list.

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