Orig Description
Score : 1400 points
Problem StatementProcess the Q queries below.
You are given two integers A_i and M_i. Determine whether there exists a positive integer K_i not exceeding 2 × 10^{18} such that A_i^{K_i} ≡ K_i (mod M_i), and find one if it exists.
Constraints
1 \leq Q \leq 100
0 \leq A_i \leq 10^9(1 \leq i \leq Q)
1 \leq M_i \leq 10^9(1 \leq i \leq Q)
InputsInput is given from Standard Input in the following format:
Q
A_1 M_1
:
A_Q M_Q
OutputsIn the i-th line, print -1 if there is no integer K_i that satisfies the condition.
Otherwise, print an integer K_i not exceeding 2 × 10^{18} such that A_i^{K_i} ≡ K_i (mod M_i). If there are multiple solutions, any of them will be accepted.
Sample Input 14
2 4
3 8
9 6
10 7
Sample Output 14
11
9
2
It can be seen that the condition is satisfied: 2^4 = 16 ≡ 4 (mod 4), 3^{11} = 177147 ≡ 11 (mod 8), 9^9 = 387420489 ≡ 9 (mod 6) and 10^2 = 100 ≡ 2 (mod 7).
Sample Input 23
177 168
2028 88772
123456789 987654321
Sample Output 27953
234831584
471523108231963269
Extracted Specification
An integer Q (1 ≤ Q ≤ 100), representing some quantity or number of queries.
A list of Q tuples (A_i, M_i), where each tuple contains two integers:
- \( A_i \) (0 ≤ \( A_i \) ≤ 10^9)
- \( M_i \) (1 ≤ \( M_i \) ≤ 10^9)
### Example Input:
```
4
2 4
3 8
9 6
10 7
```
### Function Signature:
Write a function f(queries) that takes in the input.
```python
def f(queries: List[Tuple[int, int]]):
'''
queries: a list of tuples where each tuple contains two integers
'''
```
Test Case Generator
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 \(M_i\).
Extract Arguments
def extract_arguments(fh):
Q = int(fh.readline().strip())
queries = []
for _ in range(Q):
A, M = map(int, fh.readline().strip().split())
queries.append((A, M))
return queries
# if __name__ == "__main__":
# input_path = sys.argv[1]
# with open(input_path, 'r') as fh:
# inputs = extract_arguments(fh)
# for A, M in inputs:
# f(A, M)