All divisors of an integer

Calculator for computing all divisors of a natural number and the count of divisors

Divisor Calculator

Divisors of a number

Calculates all divisors of a natural number n by systematic checking and determines the number of divisors

What are divisors?

The divisors of a number are all numbers that divide it without remainder. Every number has at least the divisors 1 and itself.

Number to analyze
Positive integer greater than 0
Note: Large numbers may require longer computation time
Examples
Example: divisors of 24
Prime factorization: 24 = 2³ × 3¹
Number of divisors: (3+1) × (1+1) = 8
Divisors: 1, 2, 3, 4, 6, 8, 12, 24
Number types
Prime 2 divisors (1, p)
Composite >2 divisors
Divisor analysis result
Number of divisors
Properties
List of all divisors
Enter a number and click "Calculate divisors"

Divisor Info

Divisor properties

Definition: d is a divisor of n if n ÷ d is an integer

Divisibility No remainder Factor

Minimum divisors: 1 and the number itself
Prime: Exactly 2 divisors

Calculation methods
1. Direct check: Test all numbers from 1 to n
2. Optimized: Test only up to √n
3. Prime factorization: Use formula
Common examples
12: 1,2,3,4,6,12 (6 divisors)
16: 1,2,4,8,16 (5 divisors)
17: 1,17 (2 divisors - prime)
20: 1,2,4,5,10,20 (6 divisors)


Methods for finding divisors

Prime factorization
\[n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}\]
\[\text{Number of divisors} = (e_1+1)(e_2+1)\ldots(e_k+1)\]

Most efficient method for large numbers

Optimized search
Test all numbers i from 1 to √n
If n mod i = 0 then i and n/i are divisors

Reduces computational effort significantly

Direct method
For every number d from 1 to n:
Check whether n mod d = 0

Simple but inefficient for large numbers

Divisibility principle
d divides n exactly when:
n = d × k for some integer k

Fundamental principle of divisibility

Step-by-step example: divisors of 24

Step 1: Prime factorization
\[\begin{aligned} 24 &= 2 \times 12 \\ &= 2 \times 2 \times 6 \\ &= 2 \times 2 \times 2 \times 3 \\ &= 2^3 \times 3^1 \end{aligned}\]

24 = 2³ × 3¹

Step 2: Number of divisors
\[\begin{aligned} \text{Number} &= (e_1 + 1)(e_2 + 1) \\ &= (3 + 1)(1 + 1) \\ &= 4 \times 2 \\ &= 8 \end{aligned}\]

24 has 8 divisors

Step 3: Determine all divisors
2⁰ × 3⁰ = 1
Smallest divisor
2¹ × 3⁰ = 2
Prime factor
2² × 3⁰ = 4
Power of 2
2³ × 3⁰ = 8
Highest 2-power
2⁰ × 3¹ = 3
Prime factor
2¹ × 3¹ = 6
Product
2² × 3¹ = 12
Half of 24
2³ × 3¹ = 24
The number itself

All divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24

General method
1. Factorize

• Decompose n into prime factors

• n = p₁^e₁ × p₂^e₂ × ...

• Determine all exponents

2. Compute count

• Use formula (e₁+1)(e₂+1)...

• Multiply all (exponent + 1)

• Result = number of divisors

3. Find divisors

• All combinations of factors

• Use different exponents

• Sort from 1 to n

Applications of divisor determination

Divisor determination is fundamental in number theory and has important practical applications:

Cryptography
  • RSA encryption relies on factorization
  • Prime tests for key generation
  • Secure hash functions
  • Digital signatures
Mathematics
  • Greatest common divisor (gcd)
  • Least common multiple (lcm)
  • Number theory and prime factorization
  • Modular arithmetic
Algorithm design
  • Optimization algorithms
  • Data structures (hash tables)
  • Complexity analysis
  • Parallelization of computations
Practical applications
  • Splitting objects into equal groups
  • Scheduling and cycles
  • Optimal packing and distribution
  • Resource allocation

Divisor determination: Fundamental number theory

The determination of all divisors of a number is a fundamental problem of number theory with wide-ranging applications. From the simple definition - a number d divides n if n divided by d has no remainder - the path leads to elegant mathematical structures and efficient algorithms. Prime factorization reveals the "DNA" of a number and enables not only the determination of all divisors, but also the computation of their count using a simple formula. These insights form the basis of modern cryptography and algorithmic optimization.

Summary

Divisor determination connects elementary arithmetic with advanced number theory. The systematic method via prime factorization transforms a seemingly simple task into an elegant mathematical procedure with precise formulas and optimized algorithms. From practical distribution problems to cryptographic security and algorithmic complexity, divisor determination shows how fundamental mathematical concepts solve complex technical challenges and enable the modern digital world.