Greatest Common Divisor

In this approach, we know the minimum value of a common factor of two integers is 1, and the maximum value of a common factor is the minimum value of the given two integers. 

Therefore we will maintain a variable ans to store GCD. We set the initial value of ans as 1. We will iterate through the integers from 2 to the minimum of X and Y. We will check for each integer if X and Y are divisible by the integer, then update the ans with the integer. In the end, we return the variable ans.

Algorithm:

• Set the variable ans as 1. The variable ans stores the greatest common divisor of X and Y.
• Iterate num from 2 to the minimum of X and Y.
  • If X and Y are divisible by num, set ans as num.
• Finally, return the variable ans.

In this approach, we can use the Euclidian theorem that states the gcd of two numbers X and Y doesn’t change if we subtract the larger number from the smaller number. Therefore we can keep subtracting the smaller number from the larger number until the larger number is greater than, the smaller number.

Instead of repeatedly subtracting the number, we can use the modulo of the larger number with the smaller number and repeat the process until one of the numbers becomes 0, then we return the other number.
 

Therefore according to the Euclidian Theorem -: 

gcd(X, Y) = gcd(Y, X % Y)
If X and Y are not equal 0.

We will create a recursive function findGcd(X, Y) that returns the greatest common divisor of the numbers X and Y.

Algorithm:

• If Y is equal to 0, then return X. This will be the base case of the recursive function.
• We will create a recursive call to find the gcd. We will return findGcd(Y, X % Y).