The algorithm you described is known as the Euclidean Algorithm for finding the greatest common divisor of two numbers. It is a well-known algorithm used in number theory and mathematics.

I can provide you with Python code that implements this algorithm recursively to find the denominator of the division that has a remainder of 0:

```python
def euclidean_algorithm(a, b):
    if b == 0:
        return a
    else:
        q = a // b
        r = a % b
        return euclidean_algorithm(b, r)

def find_denominator(a, b):
    if a < 0 or b < 0:
        print("Both numbers should be natural numbers.")
        return None
    
    x = max(a, b)
    y = min(a, b)

    if y == 0:
        return None

    return euclidean_algorithm(x, y)

# Example usage
a = 63
b = 27
denominator = find_denominator(a, b)
print(f"The denominator of the division with remainder 0 is: {denominator}")
```

You can use this code by providing two natural numbers, and it will return the denominator of the division that has a remainder of 0. Feel free to adjust the input values or integrate this code into your project as needed.