Sure! Below is a Python function that generates test cases for the given problem statement "Divisor".

```python
import random

def tcgen():
    # Generate a random value for N within the given constraint
    N = random.randint(1, 12)
    
    return N

# Example usage
for _ in range(5):  # Generate 5 test cases
    print(tcgen())
```

This function `tcgen()` generates a random integer \(N\) within the range [1, 12], as specified in the problem constraints. Each call to `tcgen()` will produce a new value for \(N\) which can be used as an input to test the solution for the problem.