Divisor Calculator

Find all divisors (factors) of any number. See factor pairs, prime factorization, divisor count, and sum of divisors.

Enter a Number

Result

Number of Divisors
24
total divisors

Step-by-Step Solution

360 = 2^3 x 3^2 x 5^1

Understanding Divisors and Factors

A divisor (or factor) of a number n is any integer that divides n evenly, leaving no remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. Finding divisors is a foundational concept in number theory and has applications across mathematics, computer science, and cryptography.

Key Concepts

Divisors (Factors)

All positive integers that divide evenly into the given number with zero remainder.

n mod d = 0

Factor Pairs

Two numbers that multiply together to produce the original number.

a x b = n

Prime Factorization

Expressing a number as a product of its prime factors raised to their powers.

n = p1^a1 x p2^a2 x ...

Number of Divisors

Given prime factorization, the count is the product of each exponent plus one.

tau(n) = (a1+1)(a2+1)...

Sum of Divisors

The sum of all positive divisors of n, denoted sigma(n).

sigma(n) = sum of all divisors

Perfect Numbers

A number where the sum of its proper divisors (excluding itself) equals the number.

sigma(n) - n = n

How to Find All Divisors

The most efficient method is to check all integers from 1 to the square root of n. For each divisor d found, both d and n/d are divisors. This significantly reduces the number of checks needed compared to testing every number up to n.

Using Prime Factorization

Once you have the prime factorization of a number, you can systematically generate all divisors by taking every combination of prime factors at each possible power level. For example, if n = 23 x 32, the divisors include 20x30=1, 21x30=2, and so on.

Practical Applications

  • Simplifying fractions by finding common divisors.
  • Determining the GCD (Greatest Common Divisor) of two numbers.
  • Solving problems in modular arithmetic and cryptography (RSA encryption).
  • Scheduling and time management (e.g., finding common intervals).
  • Number theory research including perfect numbers and amicable numbers.

Special Types of Numbers

  • Prime numbers have exactly two divisors: 1 and the number itself.
  • Perfect numbers equal the sum of their proper divisors (e.g., 6 = 1+2+3).
  • Abundant numbers have a proper divisor sum greater than the number.
  • Deficient numbers have a proper divisor sum less than the number.
  • Highly composite numbers have more divisors than any smaller positive integer.