Common divisors of two integers

Calculator for computing all common divisors of two natural numbers

Common Divisors Calculator

Find common divisors

Finds all common divisors of two natural numbers and computes the greatest common divisor (gcd) as well as the number of divisors.

Common divisors of 100 and 400
Enter numbers
Natural number greater than 0
Natural number greater than 0
Calculation result
Number of common divisors:
List of common divisors
Calculation: All numbers that divide both input values without remainder

Divisors Info

Properties

Common divisors: Numbers that divide both values without remainder

gcd Divisors Factors

Definition: A number d is a common divisor of a and b if a ÷ d and b ÷ d both have no remainder
gcd: The greatest common divisor

Examples
12 and 18: Common divisors: 1, 2, 3, 6
15 and 25: Common divisors: 1, 5
7 and 11: Common divisors: 1 (primes)

Theory of common divisors

Definition of divisor
A number d divides a if a = d × k for some integer k

Mathematical definition of divisibility

Greatest common divisor (gcd)
\[\gcd(a,b) = \max\{d : d|a \text{ and } d|b\}\]

The greatest among all common divisors

Euclidean algorithm
\[\gcd(a,b) = \gcd(b, a \bmod b)\]

Efficient algorithm for gcd computation

Properties
\[\gcd(a,b) = \gcd(|a|,|b|)\] \[\gcd(a,0) = |a|\]

Important mathematical properties

Examples of common divisors

Example 1: 100 and 400
Divisors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
Divisors of 400: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400
Common divisors:
1, 2, 4, 5, 10, 20, 25, 50, 100
Count: 9
gcd: 100
Example 2: 12 and 18
Divisors of 12: 1, 2, 3, 4, 6, 12
Divisors of 18: 1, 2, 3, 6, 9, 18
Common divisors:
1, 2, 3, 6
Count: 4
gcd: 6
Steps to determine common divisors
1. Find all divisors of a
2. Find all divisors of b
3. Take intersection
4. Determine the greatest divisor

Systematic approach

Applications of common divisors

Common divisors are fundamental in number theory and have practical applications:

Fraction arithmetic
  • Reduce and expand fractions
  • Find common denominators
  • Simplify fractions
  • Determine equivalent fractions
Cryptography
  • RSA encryption
  • Key generation
  • Prime factorization
  • Security analysis
Engineering
  • Gear ratios
  • Periodic systems
  • Optimization of ratios
  • Modular arithmetic
Mathematics education
  • Foundations of number theory
  • Algorithmic thinking
  • Problem solving strategies
  • Pattern recognition

Common divisors: Fundamentals of number theory

Common divisors are a fundamental concept of elementary number theory. The divisors of a number are all numbers that divide it without remainder. Common divisors of two numbers are the numbers that divide both original numbers without remainder. The greatest common divisor (gcd) plays a central role in many mathematical applications and is closely related to concepts like prime factorization and modular arithmetic.

Properties
  • Every number has at least two divisors: 1 and itself
  • The gcd is the greatest common divisor
  • All common divisors divide the gcd
  • Prime numbers have only themselves and 1 as divisors
Importance
  • Basis for fraction arithmetic
  • Important in cryptography
  • Application in algorithms
  • Foundation for further number theory
Computation methods
  • Complete divisor lists
  • Euclidean algorithm
  • Prime factorization
  • Modular arithmetic
Summary

Common divisors link basic arithmetic operations with deep mathematical concepts. From simple division to complex number theory they bridge elementary and advanced mathematics. Their applications range from fraction arithmetic to modern cryptography and demonstrate how fundamental mathematical concepts underpin complex technical applications.