GCD/LCM Calculator

GCD is the largest positive integer that divides all given numbers without leaving a remainder. It's also known as HCF (Highest Common Factor). For example, GCD(12, 18) = 6 because 6 is the largest number that divides both 12 and 18 evenly.

LCM is the smallest positive integer that is divisible by all given numbers. For example, LCM(4, 6) = 12 because 12 is the smallest number that both 4 and 6 can divide into without remainder.

The Euclidean algorithm finds GCD by repeatedly applying the division algorithm: divide the larger number by the smaller, replace the larger with the smaller and the smaller with the remainder, and repeat until the remainder is zero. The last non-zero remainder is the GCD. For example, to find GCD(48, 18): 48÷18=2 remainder 12, then 18÷12=1 remainder 6, then 12÷6=2 remainder 0, so GCD=6.

GCD and LCM are related by the formula: LCM(a,b) = |a×b| / GCD(a,b). This means if you know the GCD of two numbers, you can easily calculate their LCM, and vice versa. This relationship holds for any two positive integers.

Yes, you can calculate GCD and LCM for three or more numbers. For GCD, use the property: GCD(a,b,c) = GCD(GCD(a,b),c). For LCM, use: LCM(a,b,c) = LCM(LCM(a,b),c). Simply enter multiple numbers separated by commas.

GCD has many applications: simplifying fractions (dividing numerator and denominator by their GCD), solving Diophantine equations, finding modular inverses in cryptography, determining if two numbers are coprime (GCD=1), and in the Euclidean algorithm which is fundamental in number theory.

LCM is used in many real-life situations: finding when events will coincide (like bus schedules), adding or subtracting fractions with different denominators, scheduling recurring tasks, solving problems involving periodic phenomena, and in music for finding rhythmic patterns.

If GCD(a,b) = 1, the numbers are called coprime or relatively prime. This means they share no common factors other than 1. For example, 8 and 15 are coprime because their only common divisor is 1. Coprime numbers are important in cryptography and number theory.

To find GCD using prime factorization: (1) Find the prime factors of each number, (2) Identify the common prime factors, (3) For each common prime factor, take the lowest power, (4) Multiply these together. For example, 36=2²×3² and 48=2⁴×3¹, so GCD=2²×3¹=12.

To find LCM using prime factorization: (1) Find the prime factors of each number, (2) For each prime factor that appears in any number, take the highest power, (3) Multiply these together. For example, 12=2²×3¹ and 18=2¹×3², so LCM=2²×3²=36.

Yes, GCD and LCM are always positive integers regardless of the sign of the input numbers. The calculator uses the absolute values of the numbers. For example, GCD(-12, 18) = 6 and LCM(-4, 6) = 12.

The Euclidean algorithm has a time complexity of O(log(min(a,b))), making it very efficient even for very large numbers. It's one of the oldest and most efficient algorithms in mathematics, dating back to ancient Greece around 300 BC.

To simplify a fraction, divide both the numerator and denominator by their GCD. For example, to simplify 24/36: GCD(24,36)=12, so 24÷12=2 and 36÷12=3, giving the simplified fraction 2/3. This ensures the fraction is in its lowest terms.

Yes, this calculator can handle large integers efficiently using the Euclidean algorithm. However, extremely large numbers (hundreds of digits) may be limited by JavaScript's number precision. For most practical purposes, it works perfectly for numbers up to 15-16 digits.