How the Algorithm Works

j25“`python
def gcd_Euclidean(a, b):
if b == 0:
return a

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return gcd_Euclidean(b, a % b)

“This code snippet calculates the greatest common divisor (GCD) between two integersaandb` using the Euclidean algorithm. Here is a simplified version of the code with comments:

def gcd_Euclidean(a, b):
    # If `b` is 0, `a` is the GCD
    if b == 0:
        return a

    # Recursively calculate the GCD of `b` and `a` % `b`
    return gcd_Euclidean(b, a % b)

1. If b is 0, it means that a is the GCD because there is no more common divisor to find.
2. If b is not 0, the algorithm calculates the remainder (a % b) when a is divided by b. The remainder represents the common divisor between a and b.
3. The algorithm then makes a recursive call to gcd_Euclidean with b and a % b. This continues until b becomes 0, at which point the algorithm returns the GCD.

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Example

Let’s calculate the GCD of 12 and 18 using the Euclidean algorithm:

1. gcd(12, 18) is called.
2. Since 18 is not 0, we calculate 12 % 18, which equals 6.
3. gcd(18, 6) is called.
4. Since 6 is not 0, we calculate 18 % 6, which equals 0.
5. We have reached a case where b (0) is 0, so gcd(18, 6) returns 6.
6. This 6 is passed back to the previous recursive call, gcd(18, 6), which also returns 6.
7. Finally, gcd(12, 18) returns 6, which is the GCD of 12 and 18.