Certainly! Below is a test case generator for the given problem statement. This generator will create random test cases based on the constraints provided.

```python
import random

def tcgen():
    Q = random.randint(1, 100)
    queries = []
    
    for _ in range(Q):
        A_i = random.randint(0, 10**9)
        M_i = random.randint(1, 10**9)
        queries.append((A_i, M_i))
    
    return Q, queries

def print_test_case(Q, queries):
    print(Q)
    for A_i, M_i in queries:
        print(A_i, M_i)

# Example usage:
if __name__ == "__main__":
    Q, queries = tcgen()
    print_test_case(Q, queries)
```

The `tcgen` function generates a random number of queries \(Q\) (between 1 and 100). For each query, it picks random values for \(A_i\) (between 0 and \(10^9\)) and \(M_i\) (between 1 and \(10^9\)). The `print_test_case` function prints the generated test case in the required format.

You can run the script to generate and print a random test case. Here is an example of how the generated output might look:

```plaintext
7
12345678 98765432
0 1
999999999 1000000000
54321 1234567
10 7
177 168
2028 88772
```

This output corresponds to 7 queries with the respective pairs of \(A_i\) and \(M_i\).