The greatest common divisor (GCD) of two integers is the largest integer that evenly divides each of the two numbers. Write a method gcd that returns the greatest common divisor of two integers. [Hint: You might want to use Euclid’s algorithm. You can find information about it at en.wikipedia.org/wiki/Euclidean_algorithm.] Incorporate the method into an application that reads two values from the user and displays the result.

Euclid’s Algorithm is a time-honored technique for finding the greatest common divisor (GCD) of two integers. It is based on the principle that the GCD of two numbers also divides their difference.

This classical method can be visualized as a process of repetitive subtraction, where you continually subtract the smaller number from the larger one until the numbers are equal, which would be the GCD. In practice, modern interpretations use division to speed up the process. For integers 'a' and 'b', with 'a' greater than 'b', you divide 'a' by 'b'. If the remainder, denoted as 'r', is zero, 'b' becomes the GCD. If not, you apply the algorithm recursively to 'b' and 'r'.

This technique simplifies the problem step by step, rapidly narrowing down the range of potential divisors, making it a highly efficient method for computing the GCD. Moreover, the simplicity of the method makes it applicable in various mathematical contexts and practical applications like simplifying fractions or cryptographic algorithms.

Recursive methods are programming techniques where a function calls itself to solve a smaller instance of the same problem. This approach can be ideal for problems that have repetitive or self-similar patterns, such as the steps in Euclid's Algorithm.

When implementing a recursive function, like the gcd method in our exercise, there are two crucial components to consider: the base case and the recursive step. The base case is the simplest instance which can be solved without further recursion—here, if 'y' is 0, then 'x' is the GCD, since any number divided by zero has a remainder of itself. The recursive step involves calling the same function with a new set of parameters drawn from the original problem, specifically 'y' and the remainder of 'x' divided by 'y'.

It is important to ensure that recursive calls will eventually reach a base case to prevent infinite recursion. For gcd, this is guaranteed since the remainder decreases with each call, eventually reaching zero.

The modulo operator '%', which calculates the remainder of division of one number by another, is pivotal in both Euclid’s Algorithm and the recursive method we've discussed.

This operator is instrumental in reducing a complex problem to a more manageable one—a key tenet of recursive methods. When 'x' is divided by 'y', the modulo operator gives us 'r', the remainder. In essence, it helps to update the values for each subsequent recursive call in the gcd method until the remainder is zero, signaling that we have found our GCD.

The modulo operator is often preferred over subtraction in finding remainders due to its efficiency, and it's widely supported in programming languages, making it a practical tool for not only the gcd computation but also in various domains where division and factoring are involved.