Python Program for Extended Euclidean algorithms

In this article, we will learn about the solution to the problem statement given below.

Problem statement − Given two numbers we need to calculate gcd of those two numbers and display them.

GCD Greatest Common Divisor of two numbers is the largest number that can divide both of them. Here we follow the euclidean approach to compute the gcd i.e. to repeatedly divide the numbers and stop when the remainder becomes zero. Here we extend the algorithm based on previous values obtained in recursion.

Now let’s observe the solution in the implementation below −

Example

# extended Euclidean Algorithm
def gcdExtended(a, b, x, y):
   # Base Case
   if a == 0 :
      x = 0
      y = 1
      return b
   x1 = 1
   y1 = 1 # storing the result
   gcd = gcdExtended(b%a, a, x1, y1)
   # Update x and y with previous calculated values
   x = y1 - (b/a) * x1
   y = x1
   return gcd
x = 1
y = 1
a = 11
b = 15
g = gcdExtended(a, b, x, y)
print("gcd of ", a , "&" , b, " is = ", g)

Output

gcd of 11 & 15 is = 1

All the variables are declared in the local scope and their references are seen in the figure above.

Conclusion

In this article, we have learned about how we can make a Python Program for Extended Euclidean algorithms