Step by Step solution about How to resolve the algorithm Greatest common divisor step by step in the Haskell programming language

The Haskell source code provided calculates the greatest common divisor (GCD) of two integers using the Euclidean algorithm. Here's a detailed breakdown:

gcd :: (Integral a) => a -> a -> a

This is the main function that calculates the GCD. It takes two arguments of type a, which must be instances of the Integral typeclass. The Integral typeclass defines operations for numeric types like integers, representing integers or fractional numbers that can be added, subtracted, multiplied, and divided.

gcd x y = gcd_ (abs x) (abs y)

The gcd function calls a helper function, gcd_, to calculate the GCD. It first takes the absolute values of the input integers x and y using abs. This ensures that the GCD calculation is performed regardless of the input integers' signs.

where

gcd_ a 0 = a

The helper function gcd_ is defined within the where block. It implements the Euclidean algorithm to find the GCD. The first equation, gcd_ a 0 = a, states that if one of the input integers is 0, the GCD is the other integer. This serves as the base case for the recursive algorithm.

gcd_ a b = gcd_ b (a rem b)

The second equation, gcd_ a b = gcd_ b (a rem b), represents the recursive step of the Euclidean algorithm. It calculates the remainder when a is divided by b using the rem operator. The result of the division is treated as the new a, and the previous b becomes the new b. The recursion continues until the second argument becomes 0, at which point the GCD is found.

In summary, this Haskell code efficiently calculates the GCD of two integers using the Euclidean algorithm. It considers the absolute values of the input integers to handle both positive and negative numbers and uses a recursive approach to find the common factors between them.

gcd :: (Integral a) => a -> a -> a
gcd x y = gcd_ (abs x) (abs y)
  where
    gcd_ a 0 = a
    gcd_ a b = gcd_ b (a `rem` b)