Certainly! Here is the test case generator for the given problem statement:

```python
import random

def tcgen():
    # Number of queries
    Q = random.randint(1, 100)
    
    queries = []
    
    for _ in range(Q):
        # Ai can be 0 or any integer up to 10^9
        A_i = random.randint(0, 10**9)
        # Mi can be any integer from 1 to 10^9
        M_i = random.randint(1, 10**9)
        queries.append((A_i, M_i))
        
    return Q, queries

# Example of how to use the generated test case
Q, queries = tcgen()
print(Q)
for A_i, M_i in queries:
    print(A_i, M_i)
```

In the test case generator:

1. `Q` is randomly chosen between 1 and 100.
2. For each query, `A_i` is randomly chosen between 0 and \(10^9\).
3. `M_i` is randomly chosen between 1 and \(10^9\). 

The function returns `Q` and the list of queries, each containing a tuple `(A_i, M_i)`.

To see how the test case generator works, you can run the `tcgen` function and print the values.