Calculator for finding the GCD and LCM of numbers.

Greatest Common Divisor (GCD) - is the largest positive integer that divides each of the given numbers without leaving a remainder.

Least Common Multiple (LCM) - is the smallest number that is divisible by all given numbers without leaving a remainder.

How to find GCD

1. Divisor enumeration method
   This method can be labor-intensive for large numbers.
   The method consists of listing all divisors of the numbers and selecting the largest common divisor.

   Example:
   Let's find GCD for numbers 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.
   Answer: GCD(12,18) = 6.

2. Prime factorization
   The essence of this method is to represent each number as a product of prime numbers.
   This method works well for small numbers and can find GCD for several numbers at once.
   Algorithm:
     Factor each number using division.
     Find all common factors — those factors that appear in each given number.
     Among them, select factors with the smallest exponent.
     Multiply them together to get the GCD.
   Example:
   GCD(12,18)

   Number		Factors		In exponential form
   12	=	2×2×3	=	22 × 31
   18	=	2×3×3	=	21 × 32

   Common factors with the smallest exponents:  2¹ * 3¹ = 6
3. Euclidean algorithm
   The most convenient method. Well suited for large numbers. But only for two numbers per calculation.
   Algorithm:
     Divide the larger number by the smaller one and find the remainder.
     Divide the smaller number by the remainder from the previous division.
     Repeat the process until the remainder becomes zero.
     The GCD is the last non-zero remainder.
     If the remainder is zero at the first division, then the GCD is the smaller number.
   Example:

   Division	Quotient	Remainder
   18÷12 =	1	6
   12÷6 =	2	0

    GCD(12,18) = 6

How to find LCM

1. Using known GCD
   The simplest and most accessible method. However, it requires knowledge of GCD and is only possible for two numbers.
   The method is based on the following formula:

   $LCM=\frac{\left|a·b\right|}{GCD}$
   where a and b are given numbers.

   Example:
   Let's find LCM for numbers 12 and 18
   First, determine GCD(12,18) = 6
   $LCM=\frac{12·18}{6}=36$
2. Prime factorization method
   The essence of the method is to factor a number into several prime numbers — factors.
   This method is suitable for any number of values.
   Example:
   Let's find LCM for numbers 12 and 18
     Factor these numbers into prime factors:

   Number		Factors		In exponential form
   12	=	2×2×3	=	22 × 31
   18	=	2×3×3	=	21 × 32

       If there are repeating factors, choose the one with the largest exponent. Such factors are highlighted in green.
       Multiplying all unique factors with the highest exponents together gives the LCM of the given numbers.

   22 × 32

   =

   36

       

At first glance, it may seem that finding GCD and LCM of numbers is a labor-intensive task. But with a little practice and checking yourself with our GCD and LCM calculator, you can master the skill of calculating GCD and LCM perfectly.
Remember that GCD is the largest number, while LCM is, conversely, the smallest.