Python Coding Test Course, Euclidean Algorithm

Hello. In this blog post, we will take a detailed look at the Euclidean algorithm, which frequently appears in algorithm exams and real employment processes, and solve coding problems utilizing this method.

1. What is the Euclidean Algorithm?

The Euclidean algorithm is an efficient method for finding the greatest common divisor (GCD) of two integers, first proposed by the ancient Greek mathematician Euclid. This method works by repeatedly dividing the two numbers to find their GCD.

Principle of the Euclidean Algorithm

For two given numbers a and b (a > b), GCD(a, b) is equal to GCD(b, a % b). This process is repeated until b becomes 0, and finally, a is the GCD.

Example

For example, let’s find the GCD of 48 and 18.

  1. GCD(48, 18) → GCD(18, 48 % 18) → GCD(18, 12)
  2. GCD(18, 12) → GCD(12, 18 % 12) → GCD(12, 6)
  3. GCD(12, 6) → GCD(6, 12 % 6) → GCD(6, 0)
  4. GCD(6, 0) = 6

2. Problem Definition

Now, let’s define an algorithm problem based on the Euclidean algorithm.

Problem: Write a program that takes two integers as input and outputs their greatest common divisor.

Input: Two integers A and B (0 < A, B < 10^9)

Output: The greatest common divisor GCD(A, B)

3. Problem Solving Process

To solve the above problem, we will go through a series of steps.

3.1 Problem Analysis

First, two integers will be given as input. The goal is to find the GCD of these two numbers. It is important to note that since the range of the input numbers is quite large, the algorithm needs to be efficient. The Euclidean algorithm has a time complexity of O(log(min(A, B))) which makes it suitable.

3.2 Algorithm Design

We will use the basic recursive approach of the Euclidean algorithm to find the GCD. Below are the main steps of the algorithm.

  1. Define a function that takes two integers as arguments.
  2. Compare the larger and smaller of the two numbers and compute the remainder when the larger number is divided by the smaller number.
  3. This process is repeated until the remainder becomes 0.
  4. When the remainder is 0, the smaller number at that time is the GCD.

3.3 Python Code Implementation

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

# Input handling
A, B = map(int, input("Please enter two integers A and B: ").split())
print("The greatest common divisor is:", gcd(A, B))

The above code uses the basic Euclidean algorithm to calculate the GCD. It takes two numbers A and B from the user, calls the gcd function, and outputs the result.

4. Complexity Analysis

The time complexity of the Euclidean algorithm is O(log(min(A, B))). This is because the two numbers are halved at each step. This algorithm is very efficient and works quickly even for large numbers.

5. Various Modifications and Applications

The Euclidean algorithm is not just limited to finding GCDs; it can also be applied to solve various other problems. For example:

  • Simplifying fractions: By taking two integers as arguments, you can divide the numerator and the denominator by their GCD to express a complete fraction.
  • Least common multiple: By dividing the product of two numbers by their GCD, you can calculate the least common multiple.

6. Conclusion

In this post, we explored the Euclidean algorithm in detail. It was a great opportunity to study the theory and write actual code through a problem that frequently appears in algorithm exams. I encourage you to use the Euclidean algorithm to solve various problems. Happy Coding!

Now, continue to learn about more Python-related algorithm problems and solutions. Mastering algorithms is an important part of job preparation, and experiencing various problems will be beneficial.