Implementing and understanding the Euclidean algorithm for GCD and its relation to LCM.
The Greatest Common Divisor (GCD) of two integers is the largest positive integer that divides both numbers without leaving a remainder. The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of both numbers. These two concepts are fundamental in number theory and have various applications in programming, such as simplifying fractions or solving problems involving periodic events. The most efficient method for calculating the GCD is the Euclidean algorithm. It's an elegant recursive algorithm based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. The modern version uses the remainder: `gcd(a, b)` is the same as `gcd(b, a % b)`, with the base case `gcd(a, 0) = a`. This algorithm is extremely fast, with a time complexity logarithmic in the size of the smaller number. Once the GCD is known, calculating the LCM is straightforward using the identity: `lcm(a, b) * gcd(a, b) = |a * b|`. Therefore, `lcm(a, b) = (|a * b|) / gcd(a, b)`. Understanding how to implement the Euclidean algorithm efficiently is a basic but essential skill for any competitive programmer or algorithm designer.