Python Program for Basic Euclidean Algorithms

Given two numbers a and b, the task is to find their greatest common divisor (GCD), which is the largest number that divides both a and b completely. For example:

Input: a = 10, b = 15 -> Output: 5
Input: a = 31, b = 2 -> Output: 1

Let’s explore different methods to find the GCD in Python.

Using math.gcd()

The math module provides a built-in gcd() function that internally implements the optimized Euclidean algorithm. This is the most efficient and pythonic way to find the GCD.

import math
a, b = 10, 15
print("GCD is", math.gcd(a, b))

GCD is 5

Explanation: math.gcd(a, b) directly returns the greatest common divisor of the two numbers.

Using Iterative Euclidean Algorithm

This approach repeatedly replaces the larger number with the remainder until one becomes zero. The non-zero number left is the GCD.

a, b = 10, 15
while b:
    a, b = b, a % b

print("GCD is", a)

GCD is 5

Explanation:

• The loop continues until b becomes zero.
• In each iteration, a is replaced with b, and b with a % b.
• Once b is zero, a holds the GCD.

Using Recursive Euclidean Algorithm

This is the traditional recursive form of the algorithm. It calls itself repeatedly by swapping a and b % a until a becomes zero.

def gcd(a, b):
    if a == 0:
        return b
    return gcd(b % a, a)

a, b = 10, 15
print("GCD is", gcd(a, b))

GCD is 5

Explanation:

• The function keeps calling itself with the remainder until the base case (a == 0) is reached.
• The value of b at that point is the greatest common divisor.

Using Subtraction-based Euclidean Algorithm

This method repeatedly subtracts the smaller number from the larger until both become equal.

a, b = 10, 15
while a != b:
    if a > b:
        a -= b
    else:
        b -= a

print("GCD is", a)

GCD is 5

Explanation:

• The larger number is reduced by the smaller in each step.
• When both numbers become equal, that value is the GCD.

Please refer complete article on Basic and Extended Euclidean algorithms for more details!