Number of Divisors
Count of divisors from prime factorization exponents.
Formula
Variables
Example
12=2²·3: (2+1)(1+1)=6
How it works
If an integer is written as a product of primes, each prime factor can appear in a divisor with any exponent from 0 up to its exponent in the original number. That gives a separate set of choices for each prime, and the total number of divisors is the product of the number of choices for each exponent. The factor (a_i + 1) appears because an exponent a_i allows exactly a_i + 1 possibilities: 0, 1, 2, ..., a_i.
When to use it
Use this when you know the prime factorization of a positive integer and want to count how many divisors it has without listing them all. It is especially useful in divisor-counting problems, number theory proofs, and questions about highly composite numbers.
Worked example, step by step
Find the number of divisors of 360. 360 = 2^3 · 3^2 · 5^1 The exponents are 3, 2, and 1. d(360) = (3 + 1)(2 + 1)(1 + 1) d(360) = 4 · 3 · 2 d(360) = 24 So, 360 has 24 positive divisors.
Common mistakes to avoid
- Forgetting to include exponent 0, which makes each factor count as a_i + 1 rather than a_i
- Using the prime values themselves instead of their exponents in the product
- Applying the formula to a number before factoring it into primes completely
Did You Know?
The number of divisors comes straight from the prime factorisation, a neat trick of exponents plus one.
Share this formula
More in Number Theory
View allEuclidean Algorithm
BasicRecursively finds the greatest common divisor.
Modular Congruence
Basica and b have the same remainder when divided by m.
Fermat's Little Theorem
AdvancedHolds when p is prime and gcd(a,p)=1.
Euler's Totient
AdvancedCounts integers up to n coprime to n.
Prime Number Theorem
AdvancedApproximates the number of primes below x.
LCM from GCD
BasicLeast common multiple via the greatest common divisor.