Divisor Calculator
List every divisor of a number, plus divisor count, divisor sum, and number classification.
Divisors
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A divisor of a number is any whole number that divides it exactly, with no remainder. This calculator goes beyond listing divisors — it computes the divisor count d(n), the divisor sum σ(n), the aliquot sum, and classifies the number as perfect, abundant, or deficient. For example, 120 has 16 divisors that sum to 360. For a plainer factor list, use the factor calculator.
Divisor or factor — which term is right?
For whole numbers, divisor and factor mean the same thing. The difference is only in how you're thinking about the operation:
- Divisor — used when dividing. In 12 ÷ 3 = 4, the number 3 is the divisor.
- Factor — used when multiplying. In 3 × 4 = 12, both 3 and 4 are factors.
Number theory tends to prefer "divisor" and uses the notation d(n) and σ(n); school arithmetic tends to prefer "factor". The list of numbers is identical either way.
The divisor count function d(n)
The divisor function d(n) — sometimes written τ(n) — tells you how many positive divisors a number has, without listing them.
The rule: take the prime factorization, add 1 to each exponent, then multiply.
Worked example — 120:
- 120 = 2³ × 3¹ × 5¹
- Exponents: 3, 1, 1
- Add 1 to each: 4, 2, 2
- Multiply: 4 × 2 × 2 = 16
So 120 has exactly 16 divisors. This works because each divisor is built by choosing a power of 2 (from 2⁰ to 2³ — four options), a power of 3 (two options), and a power of 5 (two options).
You can get any prime factorization from the prime factorization calculator.
The divisor sum function σ(n)
The divisor sum σ(n) adds every divisor together, including the number itself. There's a formula that avoids adding them one by one.
For each prime power pe in the factorization, the contribution is (pe+1 − 1) ÷ (p − 1). Multiply all the contributions.
Worked example — 12 = 2² × 3:
- For 2²: (2³ − 1) ÷ (2 − 1) = 7 ÷ 1 = 7
- For 3¹: (3² − 1) ÷ (3 − 1) = 8 ÷ 2 = 4
- σ(12) = 7 × 4 = 28
Checking by hand: 1 + 2 + 3 + 4 + 6 + 12 = 28. The formula matches.
Perfect, abundant, and deficient numbers
Comparing a number to the sum of its proper divisors (all divisors except the number itself) gives a classification used throughout number theory.
| Type | Condition | Examples |
|---|---|---|
| Perfect | Proper divisors sum to exactly n | 6, 28, 496, 8128 |
| Abundant | Proper divisors sum to more than n | 12, 18, 20, 24, 30 |
| Deficient | Proper divisors sum to less than n | All primes, plus 4, 8, 9, 10 |
Perfect numbers are strikingly rare. Only 51 are known, the smallest being 6 (1 + 2 + 3 = 6) and 28 (1 + 2 + 4 + 7 + 14 = 28). Every one discovered so far is even, and whether an odd perfect number exists is still an open problem in mathematics.
Why perfect squares have an odd divisor count
Divisors normally come in pairs. For 12: (1, 12), (2, 6), (3, 4) — six divisors from three pairs. Because pairs come in twos, most numbers have an even divisor count.
Perfect squares break this pattern. For 36 the pairs are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). That last pair uses 6 twice, so it contributes only one divisor. The total is 9 — odd.
This makes a neat test: a number has an odd number of divisors if and only if it is a perfect square.
Related tools
To find divisors shared between numbers, use the common factors calculator or the GCF calculator. For multiples rather than divisors, see the LCM calculator.