Number of Divisors Calculator
The number of divisors of a whole number counts how many positive integers divide it evenly, including one and the number itself. This calculator takes a positive whole number and returns that count. It works by factoring the number into primes, then multiplying together one plus each prime's exponent, which is the standard divisor-counting function from number theory. The reason it works is neat: any divisor is built by choosing how many of each prime factor to include, from none up to the full power available, so the choices multiply. For example, sixty factors as two squared times three times five, giving three choices for the power of two, two for three and two for five, a total of twelve divisors. The calculator also lists the divisors so you can see them. Mathematicians, cryptographers, students and puzzle enthusiasts use the divisor count to study the structure of numbers, identify primes, which have exactly two divisors, and find highly composite numbers. Enter a whole number to get the count immediately; the input is a plain integer count, so no decimal formatting applies. Every figure here is computed deterministically from the rule shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
The divisor count is the product of (each prime exponent + 1). For 60 = 2^2 x 3 x 5 that is 3 x 2 x 2 = 12 divisors.
Number of Divisors formula
d(n) = (e1 + 1)(e2 + 1) ... (ek + 1)
n = p1^e1 x p2^e2 x ... x pk^ek
e1, e2, ... = prime exponents
d(n) = count of positive divisors
includes 1 and n itself
Factor the number into primes. Each divisor uses each prime to a power from zero up to its exponent, so the number of divisors is the product of one plus each exponent.
Worked example
Count the divisors of 60.
- Factor: 60 = 2^2 x 3^1 x 5^1
- Add one to each exponent: 3, 2, 2
- Multiply: 3 x 2 x 2 = 12 divisors
60 has 12 divisors. This is the calculator's default input, so the result above matches the widget exactly.
Divisor counts for sample numbers
d(n) from the prime factorization.
| n | Factorization | Divisors |
|---|---|---|
| 12 | 2^2 x 3 | 6 |
| 60 | 2^2 x 3 x 5 | 12 |
| 36 | 2^2 x 3^2 | 9 |
| 100 | 2^2 x 5^2 | 9 |
| 48 | 2^4 x 3 | 10 |
Number-theory reference: US National Institute of Standards and Technology (NIST).
Number of Divisors Calculator: frequently asked questions
What counts as a divisor?
A divisor of a number is any positive integer that divides it with no remainder. The count includes both one and the number itself. For example, the divisors of 12 are 1, 2, 3, 4, 6 and 12, so 12 has six divisors.
How does the prime-factor method work?
Write the number as a product of prime powers. Any divisor is formed by choosing each prime to a power between zero and its exponent, so the number of choices for each prime is the exponent plus one. Multiplying those choices gives the total divisor count.
How many divisors does a prime have?
Exactly two: one and itself. A prime number has no other factors, so its divisor count is always two. This is a quick way to test whether a number is prime using the divisor count.
What is a highly composite number?
A highly composite number has more divisors than any smaller positive integer, such as 12, 24, 36 and 60. They are useful in design and scheduling because they divide evenly in many ways, which is why measures like 60 minutes and 360 degrees are common.
What is the divisor-count formula?
Factor the number into primes and multiply one plus each exponent: d(n) = (e1 + 1)(e2 + 1) and so on. Since 60 is 2 squared times 3 times 5, the count is 3 times 2 times 2, which is 12.
Official sources
- Mathematical functions and integer-sequence reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.