Certainly! Here's a test case generator for the described "Divisor" problem:

```python
import random

def tcgen():
    N = random.randint(1, 12)
    return N

# Example usage:
for _ in range(10):
    print(tcgen())
```

This function will generate a random test case where `N` is a natural number between 1 and 12 inclusive. Each time you call `tcgen()`, it will return a single integer `N` which can be used as input for the problem.

If you need to generate multiple test cases at once, you can adjust the function to return a list of test cases:

```python
def generate_multiple_test_cases(num_cases):
    return [tcgen() for _ in range(num_cases)]

# Example usage:
test_cases = generate_multiple_test_cases(10)
for case in test_cases:
    print(case)
```

This code will generate and print 10 test cases for the problem. Adjust the number in `generate_multiple_test_cases(10)` to generate as many test cases as you need.