How to use Python to implement the algorithm for finding the greatest common divisor?

How to use Python to implement the algorithm for finding the greatest common divisor?

The greatest common divisor, also known as the greatest common factor, refers to the largest number among the divisors shared by two or more numbers. Calculating the greatest common divisor is a very common task in mathematics and computer fields. Python, as a popular programming language, provides a variety of methods to implement this algorithm.

The following will introduce three commonly used algorithms for implementing the greatest common divisor in Python, namely the exhaustive method, the euclidean division method, and the phase-changing subtraction method.

1. Exhaustive method
   The exhaustive method is the most intuitive but less efficient method. This method tries all possible factors one by one to find the greatest common divisor.

def gcd_exhaustive(a, b):
    if a > b:
        smaller = b
    else:
        smaller = a
    for i in range(1, smaller+1):
        if ((a % i == 0) and (b % i == 0)):
            gcd = i
    return gcd

2. Euclidean division method
   Euclidean division method, also known as Euclidean algorithm, is a recursive algorithm of Euclidean division. This algorithm is based on the following theorem: the greatest common divisor of two positive integers a and b (a > b) is equal to the greatest common divisor between the remainder c and b of a divided by b.

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

3. Additional phase subtraction method
   Additional phase subtraction method is also a recursive algorithm, which solves the greatest common divisor by continuously subtracting the difference between two numbers. However, this algorithm is less efficient and may timeout when processing large numbers.

def gcd_subtraction(a, b):
    if a == b:
        return a
    elif a > b:
        return gcd_subtraction(a-b, b)
    else:
        return gcd_subtraction(a, b-a)

can be tested by the following code:

a = 374
b = 256

print("穷举法求解最大公约数：")
print(gcd_exhaustive(a, b))

print("辗转相除法求解最大公约数：")
print(gcd_euclidean(a, b))

print("更相减损法求解最大公约数：")
print(gcd_subtraction(a, b))

According to the above code, when the input a is 374 and b is 256, the calculated greatest common divisor is 2 (using Exhaustive method), 2 (using euclidean division method) and 2 (using replacement subtraction method).

The above are three commonly used algorithms for solving the greatest common divisor using Python. Depending on the specific situation and data size, an appropriate algorithm can be selected to solve the greatest common divisor.

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