Factor Calculator
Enter any whole number and the calculator lists every positive factor (divisor), its prime factorisation, all of its factor pairs, and σ(n) — the sum of its divisors — along with whether the number is prime, composite, or perfect.
Number of factors
12
- All factors
- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Prime factorisation
- 2^2 × 3 × 5
- Factor pairs
- 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10
- Sum of divisors σ(n)
- 168
60 has 12 positive factors. Classification: Composite.
How to use this calculator
Type a single whole number into the box and the calculator updates immediately. Commas, spaces, and underscores in the input are ignored, so 1,234,567 and 1_234_567 are both accepted. Negative numbers are treated as their absolute value, since factors are conventionally non-negative. The primary result is the number of positive factors; the breakdown lists every factor in ascending order, the prime factorisation in canonical form (smallest prime first, exponents written as p^k), every factor pair, and σ(n), the sum of all positive divisors. The explanation flags whether the input is prime, composite, or perfect.
How the calculation works
A factor (or divisor) of n is any positive integer d such that n divided by d leaves no remainder. The calculator finds them by trial division: test every integer i from 1 up to √n; whenever i divides n exactly, both i and n / i are factors. This is the standard textbook algorithm (Hardy & Wright, An Introduction to the Theory of Numbers, §1.2) and runs in O(√n) time, which is fast enough for any number that fits in JavaScript's safe-integer range. The prime factorisation is built by repeatedly dividing out the smallest prime that still divides n until what remains is 1. By the fundamental theorem of arithmetic (Euclid, Elements VII), this representation is unique up to ordering. σ(n) — the sum of divisors — is computed multiplicatively: for a prime power p^k it equals (p^(k+1) − 1) / (p − 1), and for any n it is the product of σ across its distinct prime factors.
Worked example
For 60: trial divide by 1, 2, 3, 4, 5, 6, 7 — all of 1, 2, 3, 4, 5, 6 divide 60, paired with 60, 30, 20, 15, 12, 10. After √60 ≈ 7.75 every new factor is the larger half of a pair we have already found, so the complete factor list is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 — twelve factors in total. The prime factorisation is 60 = 2² × 3 × 5, since 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. The factor pairs are 1×60, 2×30, 3×20, 4×15, 5×12, 6×10. σ(60) = (2³ − 1)/(2 − 1) × (3² − 1)/(3 − 1) × (5² − 1)/(5 − 1) = 7 × 4 × 6 = 168. Since the proper-divisor sum is 168 − 60 = 108 ≠ 60, the number is not perfect. Contrast 28: factors 1, 2, 4, 7, 14, 28 sum to 56, so 28 is perfect (1 + 2 + 4 + 7 + 14 = 28).
Frequently asked questions
What is the difference between a factor, a divisor, and a multiple?
A factor and a divisor of n are the same thing: a positive integer that divides n exactly. So 1, 2, 3, 4, 6, 12 are both the factors and the divisors of 12. A multiple of n is a number you get by multiplying n by some positive integer — the multiples of 12 are 12, 24, 36, 48, 60, …. Factors look downward from n; multiples look upward. Every positive integer is a factor of itself and of every one of its multiples.
What is a prime number?
A prime number is a whole number greater than 1 whose only positive factors are 1 and itself — exactly two divisors. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Every other whole number greater than 1 is composite: it has at least one factor strictly between 1 and itself, so its divisor count is at least three. The number 1 is neither prime nor composite (it has only one divisor), and 0 is treated separately because every non-zero integer divides 0. The calculator labels these cases automatically: prime, composite, unit (1), or zero.
Why is the prime factorisation unique?
This is the fundamental theorem of arithmetic, proved in Euclid's Elements (Book VII, Propositions 30 and 32): every integer greater than 1 can be written as a product of primes in exactly one way, up to the order of the factors. For example, 60 = 2 × 2 × 3 × 5, and no other multiset of primes multiplies to 60. The calculator displays this in canonical form — smallest prime first, repeated primes collected as p^k — so 60 appears as 2² × 3 × 5. Uniqueness is what lets you compute GCD, LCM, σ(n), and Euler's φ(n) directly from the factorisation without ambiguity.
How many factors does a number have?
If n has the prime factorisation p₁^a₁ × p₂^a₂ × … × p_k^a_k, then the number of positive divisors — written d(n) or τ(n) — is exactly (a₁ + 1)(a₂ + 1) … (a_k + 1). Each exponent in a divisor can range from 0 up to a_i independently, giving that product. Example: 60 = 2² × 3 × 5 has (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12 divisors. A square number always has an odd number of divisors, since one of the factor pairs is i × i; every other number has an even count.
What is a perfect number?
A perfect number equals the sum of its proper divisors (every divisor except itself). The first few are 6 (= 1 + 2 + 3), 28 (= 1 + 2 + 4 + 7 + 14), 496, and 8128. Euclid proved that if 2^p − 1 is a Mersenne prime, then 2^(p−1)(2^p − 1) is perfect; Euler proved every even perfect number has this form. No odd perfect number is known, and none can exist below 10^1500 — one of the oldest unsolved problems in mathematics. The calculator flags perfect numbers automatically based on σ(n) − n = n.
What is σ(n), the sum of divisors?
σ(n) is the sum of every positive divisor of n, including 1 and n itself. For 12, σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. The function σ is multiplicative: σ(mn) = σ(m)σ(n) whenever m and n are coprime, which makes it cheap to compute from the prime factorisation. σ(n) appears in many classical results — perfect numbers (σ(n) = 2n), abundant numbers (σ(n) > 2n, like 12), and deficient numbers (σ(n) < 2n, like every prime). The aliquot sum σ(n) − n is the basis of the perfect / abundant / deficient classification.