The algorithm described in your request is known as the Euclidean Algorithm. It's a method for finding the greatest common divisor (GCD) of two integers and also for finding integer solutions to the equation ax + by = gcd(a, b). The process of iteratively finding remainders until reaching a remainder of 0 is a key part of the Euclidean Algorithm.

Here is a Python implementation of the Euclidean Algorithm that finds 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
        if r == 0:
            return b
        else:
            return euclidean_algorithm(b, r)

a = int(input("Enter the first natural number (a): "))
b = int(input("Enter the second natural number (b): "))

if a < 1 or b < 1:
    print("Please enter natural numbers.")
else:
    x = max(a, b)
    y = min(a, b)
    gcd = euclidean_algorithm(x, y)
    print(f"The denominator of the division that has a remainder of 0 is: {gcd}")
```

You can run this code in a Python environment to find the denominator of the division that has a remainder of 0 for two given natural numbers "a" and "b". Let me know if you need any further clarification or assistance.