GCD & LCM Calculator
Find the GCD and LCM of two or more numbers.
What Is the GCD & LCM Calculator?
This calculator finds the greatest common divisor (GCD) and least common multiple (LCM) of two or more integers using the efficient Euclidean algorithm. The GCD is the largest number that divides all input values evenly. The LCM is the smallest number that is a multiple of all input values.
Formula
How to Use
Enter two or more integers separated by commas or spaces. Click Calculate to find both the GCD and LCM with step-by-step workings.
Example Calculation
For 12, 18, 24: GCD(12, 18) = 6, GCD(6, 24) = 6. LCM(12, 18) = 36, LCM(36, 24) = 72. So GCD = 6 and LCM = 72.
Understanding GCD & LCM
The GCD and LCM are foundational concepts in number theory with wide practical applications. The GCD helps simplify fractions to their lowest terms, while the LCM is essential for adding fractions with different denominators.
The Euclidean algorithm, dating back over 2,300 years, is one of the oldest known algorithms still in widespread use. It efficiently computes the GCD by repeatedly applying the division algorithm until the remainder is zero.
Beyond basic arithmetic, GCD and LCM appear in cryptography (RSA encryption relies on properties of prime factorization), computer science (scheduling algorithms), music theory (finding common rhythmic patterns), and engineering (gear ratio calculations).
Frequently Asked Questions
What is the Euclidean algorithm?
It is an efficient method for computing GCD by repeatedly replacing the larger number with the remainder of dividing the two numbers until one becomes zero. The other number is the GCD.
How are GCD and LCM related?
For two numbers a and b: GCD(a,b) × LCM(a,b) = |a × b|. This identity allows computing LCM efficiently once GCD is known.
What are practical uses for GCD and LCM?
GCD is used to simplify fractions. LCM is used to find common denominators, schedule repeating events, and solve problems involving cycles.