**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
n = int(input(""))
for i in range(n):
    n2=int(input(""))
    L=[1]*n2
    for j in range(n2):
        for k in range(n2):
            if (k+1)%(j+1)==0:
                L[k]=L[k]*(j+1)
    s=str(L[0])
    for i in range(1,len(L)):
        s+=' '+str(L[i])
    print(s)

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that can highlight the issues in the algorithm. The buggy code has a nested loop that attempts to modify the array elements, but it does so in a way that may not always produce a valid beautiful array.

### Key Points to Consider:
1. **Edge Cases:**
   - Small values of `n` (e.g., 1, 2, 3).
   - Values of `n` that are prime numbers (e.g., 5, 7).
   - Values of `n` that are composite numbers (e.g., 4, 6, 8).

2. **Behavior of the Buggy Code:**
   - The code initializes an array `L` with all elements set to 1.
   - It then iterates over each element and multiplies it by the index `j+1` if the index `k+1` is a multiple of `j+1`.
   - This can lead to large numbers and may not always ensure that the sum of elements at positions divisible by `k` is a multiple of `k`.

### Challenging Test Input:
To expose the bug, we can use a small value of `n` that is composite, such as `6`, and include a few test cases to cover different scenarios.

```plaintext
3
1
2
6
```

### Explanation:
1. **Test Case 1: `n = 1`**
   - The array should be `[1]`, which is trivially beautiful since the sum of elements (1) is divisible by 1.
   - This tests the simplest case.

2. **Test Case 2: `n = 2`**
   - The array should be `[1, 2]`, which is beautiful since the sum of elements (3) is divisible by 1, and the second element (2) is divisible by 2.
   - This tests a small composite number.

3. **Test Case 3: `n = 6`**
   - This is the most challenging case. The array should be constructed such that the sum of elements at positions divisible by `k` is a multiple of `k` for all `k` from 1 to 6.
   - The buggy code might fail to produce a valid array for this case due to the nested loop and the way it modifies the array elements.

This test input should help expose the issues in the buggy code and ensure that the algorithm is tested thoroughly.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
def find_beautiful_array(n):
  a = []
  for k in range(1, n + 1):
    m = max(a) + 1 if a else 1
    while m * k <= 10**5:
      a.extend([m] * k)
      m += 1
  return a[:n]  # Return only the first n elements

t = int(input())
for _ in range(t):
  n = int(input())
  a = find_beautiful_array(n)
  print(*a)  # Print elements separated by space

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases or scenarios where the code is likely to fail. The main issue in the buggy code is that it incorrectly constructs the array `a` by repeatedly adding multiples of `k` until it exceeds a certain limit, which can lead to an array that does not satisfy the conditions for being beautiful.

### Step-by-Step Reasoning:

1. **Identify the Bug**: The bug in the code is that it does not ensure that the sum of elements at indices divisible by `k` is actually divisible by `k`. It simply adds multiples of `k` until a certain limit, which may not always result in a valid beautiful array.

2. **Edge Cases**:
   - **Small `n`**: For small values of `n`, the code might still produce a valid array by chance, but we can still test these cases.
   - **Large `n`**: For larger values of `n`, the code is more likely to produce an invalid array because it does not ensure the divisibility condition.

3. **Test Input**:
   - We should test with a value of `n` that is large enough to expose the bug but small enough to be manageable.
   - We should also test with multiple test cases to increase the likelihood of catching the bug.

### Chosen Test Input:
- **Test Case 1**: `n = 10`
- **Test Case 2**: `n = 50`
- **Test Case 3**: `n = 100`

### Output:
```plaintext
3
10
50
100
```

This test input will help expose the bug in the buggy code by requiring it to generate arrays of different sizes, including a large size where the incorrect construction method is more likely to fail.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;

public class Main{

    // Helper to calculate the greatest common divisor (GCD)
    private static int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }

    // Helper to calculate the least common multiple (LCM)
    private static int lcm(int a, int b) {
        return (a * b) / gcd(a, b);
    }

    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        int t = Integer.parseInt(br.readLine().trim());

        StringBuilder result = new StringBuilder();

        for (int test = 0; test < t; test++) {
            int n = Integer.parseInt(br.readLine().trim());
            int[] lcmArray = new int[n + 1];

            // Compute the LCM array
            lcmArray[1] = 1;
            for (int i = 2; i <= n; i++) {
                lcmArray[i] = lcm(lcmArray[i - 1], i);
            }

            // Prepare the result array for the current test case
            for (int i = 1; i <= n; i++) {
                result.append(lcmArray[i]).append(" ");
            }
            result.append("\n");
        }

        System.out.print(result.toString());
    }
}

// 95JH1xK38EkeDlNMRiL

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on the logic that constructs the array. The code computes the least common multiple (LCCM) of the numbers from 1 to n and uses this LCM to form the array. The issue with this approach is that it does not ensure that the sum of elements at indices that are multiples of k is divisible by k for all k from 1 to n.

To create a test case that exposes this bug, we need to find a small n where the LCM-based approach fails to meet the problem's requirements. Let's consider n = 4:

1. **n = 4:**
   - The LCM of 1, 2, 3, and 4 is 12.
   - The array generated by the buggy code would be: 1, 2, 6, 12.

   Let's check the sums:
   - For k = 1: Sum = 1 + 2 + 6 + 12 = 21 (not divisible by 1)
   - For k = 2: Sum = 2 + 12 = 14 (not divisible by 2)
   - For k = 3: Sum = 6 (divisible by 3)
   - For k = 4: Sum = 12 (divisible by 4)

   The sums for k = 1 and k = 2 are not divisible by their respective k values, which means the array is not beautiful.

Thus, a small-scale test input that exposes the bug is:

```plaintext
1
4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import java.util.Scanner;

public class Main{
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int t = scanner.nextInt(); // Number of test cases

        while (t-- > 0) {
            int n = scanner.nextInt(); // Size of the array
            int[] array = new int[n];

            // Fill the array with a value that is a multiple of all numbers from 1 to 100
            int value = 2520; // LCM of numbers from 1 to 100
            for (int i = 0; i < n; i++) {
                array[i] = value;
            }

            // Print the array
            for (int i = 0; i < n; i++) {
                System.out.print(array[i] + " ");
            }
            System.out.println();
        }

        scanner.close();
    }
}

// 9vVSU6e8a0cLQHFqpbu

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To create a challenging test input that exposes the bug in the given buggy code, let's consider the following points:

1. **Understanding the Bug**: The buggy code fills the array with the same value (2520), which is the least common multiple (LCCM) of numbers from 1 to 100. This ensures that the sum of any subset of the array will be divisible by any number from 1 to 100. However, this approach is overly simplistic and does not consider the constraints of the problem, especially for smaller arrays.

2. **Edge Cases**:
   - **Small Array Size**: For small array sizes (e.g., 1, 2, 3), the sum of the elements must still be divisible by the respective indices.
   - **Single Element Array**: An array with a single element must be divisible by 1.
   - **Array with Multiple Elements**: For arrays with multiple elements, the sum of elements at indices divisible by a given number must be divisible by that number.

3. **Challenging Input**:
   - **Test Case 1**: An array of size 1. The only element must be divisible by 1.
   - **Test Case 2**: An array of size 2. The sum of the elements must be divisible by 1, and each element must be divisible by its index.
   - **Test Case 3**: An array of size 3. The sum of the elements must be divisible by 1, the sum of elements at indices 1 and 2 must be divisible by 2, and the sum of elements at indices 1 and 3 must be divisible by 3.

Given these points, a small-scale test input that can expose the bug in the buggy code is:

```plaintext
3
1
2
3
```

This input will force the code to handle arrays of different sizes, including the smallest possible sizes, and will check if the sums of the elements meet the required divisibility conditions.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
int main() {
    cin.tie(0);
    iostream::sync_with_stdio(0);
    int t; cin >> t;
    while (t--) {
        int n; cin >> n;
        vector<int> arr(n, 1);
        for (int k = 2; k <= n; k++) {
            for (int i = k-1; i < n; i += k) {
                arr[i] *= k;
            }
        }
        for (int a : arr) cout << a << ' ';
        cout << '\n';
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios where the code's logic might fail. The buggy code initializes an array with all elements set to 1 and then multiplies the elements at indices that are multiples of `k` by `k`. This approach might not always ensure that the sum of elements at indices divisible by `k` is divisible by `k`.

Let's consider the following edge cases:
1. Small values of `n` where the array size is minimal.
2. Values of `n` that are prime, which might expose issues with the multiplication logic.
3. Values of `n` that are powers of small integers, which might cause repeated multiplications.

Given these considerations, a small-scale test input that includes these edge cases would be:

```plaintext
5
1
2
3
4
5
```

This input includes:
- `n = 1`: The smallest possible array.
- `n = 2`: A small even number.
- `n = 3`: A small prime number.
- `n = 4`: A small power of 2.
- `n = 5`: Another small prime number.

These values will help test the code's behavior in different scenarios and are likely to expose the bug in the multiplication logic.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;

#define ll long long
#define pii pair<int, int>
#define vi vector<int>
#define vl vector<ll>
#define vin(a) for(int i = 0; i < (a).size(); ++i) cin >> a[i];
#define vout(a) for(int i = 0; i < (a).size(); ++i) cout << a[i] << ' ';
#define mii map<int, int>
#define mll map<ll, ll>
#define mci map<char,int>
#define si set<int>
#define sc set<char>
#define sl set<ll>
#define pb push_back
#define eb emplace_back
#define endl "\n"
#define yes cout<<"YES"<<endl;
#define no cout<<"NO"<<endl;
#define bob cout<<"Bob"<<endl;
#define Alice cout<<"Alice"<<endl;
#define M 1000000007
#define PI 3.1415926535897932384626433832795

ll gcd(ll a,ll b) { if (b==0) return a; return gcd(b, a%b); }
ll lcm(ll a,ll b) { return a/gcd(a,b)*b; }
string to_upper(string a) { for (int i=0;i<(int)a.size();++i) if (a[i]>='a' && a[i]<='z') a[i]-='a'-'A'; return a; }
string to_lower(string a) { for (int i=0;i<(int)a.size();++i) if (a[i]>='A' && a[i]<='Z') a[i]+='a'-'A'; return a; }
bool prime(ll a) { if (a==1) return 0; for (int i=2;i<=round(sqrt(a));++i) if (a%i==0) return 0; return 1; }
bool palindrome(string s) { int n = s.length(); for (int i = 0; i < n / 2; i++) if (s[i] != s[n - 1 - i]) return 0; return 1; }
ll nCr(int n, int r) { long long res = 1;for (int i = 0; i < r; i++) {res = res * (n - i);res = res / (i + 1); }return res;}
ll factorial(ll n) {ll result = 1;for (ll i = 1; i <= n; ++i) {result *= i;}return result;}

void solve()
{
   ll n;
   cin>>n;
   vl a(n,1);
   for(int i=1;i<n;i++)
   {
      int j=i;
      while(j<n)
      {
         a[j]=a[j]*(i+1);
         j+=(i+1);
      }
   }
   vout(a);
   cout<<endl;
}

signed main()
{
   int T;
   cin >> T;
   while(T--)
   {
      solve();
   }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided buggy code, we need to focus on edge cases and scenarios where the code might fail. The key issue in the code is how it constructs the array to ensure that the sum of elements at indices divisible by \( k \) is also divisible by \( k \).

### Step-by-Step Analysis:

1. **Understanding the Bug:**
   - The code initializes an array `a` with all elements set to 1.
   - It then iterates over each index `i` and multiplies the elements at indices `j` that are multiples of `i+1` by `i+1`.
   - This approach might not correctly ensure that the sum of elements at indices divisible by \( k \) is divisible by \( k \).

2. **Edge Cases to Consider:**
   - **Small \( n \):** When \( n \) is very small, the behavior of the code can be more predictable but still needs to be checked.
   - **Special \( n \):** Values of \( n \) that are prime or have specific properties might expose the bug.
   - **Large \( n \):** Although the problem constraints limit \( n \) to 100, we should still consider larger values within this range to see if the bug becomes more apparent.

3. **Constructing the Test Input:**
   - We will start with a small \( n \) and then move to a larger \( n \) to see if the bug is more evident.
   - We will also consider a value of \( n \) that is a prime number to see if the code handles it correctly.

### Test Input:

Let's start with \( n = 3 \) and \( n = 7 \) to see if the code behaves correctly for small and prime values of \( n \).

```plaintext
2
3
7
```

### Explanation:
- **Test Case 1: \( n = 3 \)**
  - The array should be such that the sum of elements at indices divisible by 1, 2, and 3 are all divisible by their respective \( k \).
  - The code should generate an array like `[1, 2, 6]` or a similar valid array.

- **Test Case 2: \( n = 7 \)**
  - The array should be such that the sum of elements at indices divisible by 1, 2, 3, 4, 5, 6, and 7 are all divisible by their respective \( k \).
  - The code should generate an array like `[1, 2, 6, 24, 120, 720, 5040]` or a similar valid array.

By providing these test cases, we can observe if the code correctly ensures that the sums are divisible by the respective \( k \) values. If the code fails for these test cases, it will help us identify the bug.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import math

def beautiful_array(t, test_cases):
    results = []
    for n in test_cases:
        array = [math.factorial(i + 1) for i in range(n)]
        results.append(array)
    return results

# Reading input
t = int(input().strip())
test_cases = []
for _ in range(t):
    n = int(input().strip())
    test_cases.append(n)

# Generating and printing the results
results = beautiful_array(t, test_cases)
for res in results:
    print(" ".join(map(str, res)))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's analyze the problem and the code:

### Problem Analysis
- The problem requires generating an array such that the sum of elements at indices that are multiples of \( k \) is divisible by \( k \) for all \( 1 \le k \le n \).
- The buggy code uses the factorial of the index to generate the array elements, which is not guaranteed to satisfy the required condition.

### Buggy Code Analysis
- The code generates an array where each element is the factorial of its index plus one.
- Factorials grow very quickly, and the sum of factorials at specific indices may not always be divisible by the required \( k \).

### Challenging Test Input
To expose the bug, we need to create a test case where the sum of factorials at specific indices is not divisible by the required \( k \). This can be achieved by choosing a small \( n \) where the factorial values are manageable but still fail the condition.

### Step-by-Step Reasoning
1. **Small \( n \)**: Choose a small \( n \) to make the factorials manageable.
2. **Check Sum Divisibility**: Ensure the sum of factorials at specific indices is not divisible by \( k \).

### Test Case
Let's choose \( n = 5 \). We will generate the array using the factorial of the index plus one and check the sums for \( k = 1, 2, 3, 4, 5 \).

- For \( n = 5 \):
  - Array: \( [1!, 2!, 3!, 4!, 5!] = [1, 2, 6, 24, 120] \)
  - Check sums:
    - \( k = 1 \): \( 1 + 2 + 6 + 24 + 120 = 153 \) (not divisible by 1)
    - \( k = 2 \): \( 2 + 24 = 26 \) (not divisible by 2)
    - \( k = 3 \): \( 6 = 6 \) (divisible by 3)
    - \( k = 4 \): \( 24 = 24 \) (divisible by 4)
    - \( k = 5 \): \( 120 = 120 \) (divisible by 5)

Since the sum for \( k = 1 \) and \( k = 2 \) are not divisible by their respective \( k \), this test case will expose the bug.

### Output
```plaintext
1
5
```

This test case will show that the buggy code does not generate a valid beautiful array for \( n = 5 \).


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
def factorial(n):
    if n == 0 or n == 1:
        return 1
    result = 1
    for i in range(2, n + 1):
        result *= i
    return result

def find_beautiful_array(n):
    n_fact = factorial(n)
    array = [(i + 1) * n_fact for i in range(n)]
    return array

def main():
    import sys
    input = sys.stdin.read
    data = input().split()

    t = int(data[0])
    results = []
    index = 1
    for _ in range(t):
        n = int(data[index])
        index += 1
        beautiful_array = find_beautiful_array(n)
        results.append(beautiful_array)

    for result in results:
        print(" ".join(map(str, result)))

if __name__ == "__main__":
    main()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on the conditions where the code might fail. The bug in the code is that it uses the factorial of `n` to construct the array, which is not necessary and can lead to very large numbers. This can cause issues with the constraints and the expected output.

### Step-by-Step Analysis:
1. **Factorial Usage**: The code uses the factorial of `n` to construct the array. This can lead to very large numbers, especially for larger values of `n`.
2. **Constraints**: The constraints are \(1 \le n \le 100\), and the output should be a valid array of integers that satisfies the conditions.
3. **Edge Cases**:
   - Small values of `n` (e.g., 1, 2, 3) to check basic functionality.
   - Values of `n` that are prime (e.g., 5, 7) to ensure the sum of indices divisible by `k` is handled correctly.
   - Values of `n` that are composite (e.g., 6, 10) to ensure the sum of indices divisible by `k` is handled correctly for multiple divisors.

### Challenging Test Input:
We will choose a small set of test cases that include both small values and values that are prime and composite to ensure the code handles all conditions correctly.

```plaintext
5
1
2
3
5
6
```

### Explanation:
- **Test Case 1 (n = 1)**: This is the smallest possible input. The array should be `[1]`, which is trivially beautiful.
- **Test Case 2 (n = 2)**: The array should be `[2, 2]` or similar, ensuring that the sum of all elements is divisible by 1 and 2.
- **Test Case 3 (n = 3)**: The array should be `[3, 3, 3]` or similar, ensuring that the sum of all elements is divisible by 1, 2, and 3.
- **Test Case 4 (n = 5)**: A prime number. The array should be `[5, 5, 5, 5, 5]` or similar, ensuring that the sum of all elements is divisible by 1, 2, 3, 4, and 5.
- **Test Case 5 (n = 6)**: A composite number. The array should be `[6, 6, 6, 6, 6, 6]` or similar, ensuring that the sum of all elements is divisible by 1, 2, 3, 4, 5, and 6.

This test input will help expose the bug in the code, as the factorial approach will produce very large numbers that are not necessary for the problem's requirements.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
public class MyClass {
    public static void main(String args[]) {
      System.out.println(4+" "+22+" "+18);
      System.out.println(10+" "+6+" "+15+" "+32+" "+125+" "+54);
      System.out.println(23+" "+18+" "+27+" "+36+" "+5+" "+66+" "+7);
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that are likely to cause issues. The buggy code provided does not actually generate the arrays dynamically but instead prints hardcoded values. This means it will fail for any input that does not match the hardcoded outputs.

### Steps to Generate the Test Input

1. **Edge Cases:**
   - Test with the minimum and maximum values for `n`.
   - Test with a single element array.
   - Test with an array where `n` is a prime number.
   - Test with an array where `n` is a composite number.

2. **Valid Test Input:**
   - The test input should be complete and valid, meaning it should follow the format specified in the problem description.

### Chosen Test Cases
- **n = 1:** Single element array.
- **n = 2:** Smallest composite number.
- **n = 3:** Smallest prime number.
- **n = 4:** Small composite number.
- **n = 100:** Maximum value for `n`.

### Test Input
```plaintext
5
1
2
3
4
100
```

### Explanation
- **n = 1:** The array should be `[x]` where `x` is divisible by 1.
- **n = 2:** The array should be `[a, b]` where `a + b` is divisible by 1 and `b` is divisible by 2.
- **n = 3:** The array should be `[a, b, c]` where `a + b + c` is divisible by 1, `b + c` is divisible by 2, and `c` is divisible by 3.
- **n = 4:** The array should be `[a, b, c, d]` where `a + b + c + d` is divisible by 1, `b + d` is divisible by 2, `c` is divisible by 3, and `d` is divisible by 4.
- **n = 100:** The array should be `[a1, a2, ..., a100]` where the sums of elements at indices divisible by each `k` (1 to 100) are divisible by `k`.

This test input will help expose the bug in the buggy code, as the hardcoded values in the code will not match the required outputs for these test cases.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import java.util.Arrays;
import java.util.Scanner;

/**
 * A_Array_Divisibility
 */
public class A_Array_Divisibility {

    public static void main(String[] args) {
        Scanner scn = new Scanner(System.in);
        int t = scn.nextInt();
        for (int i = 0; i < t; i++) {
            int n = scn.nextInt();
            int arr[] = new int[n + 1];
            Arrays.fill(arr, 1);
            for (int j = 1; j <= n; j++) {
                for (int k = j; k <= n; k += j) {
                    arr[k] *=k;
                }
            }
            for (int j = 1; j <= n; j++) {
                System.out.print(arr[j] + " ");
            }
            System.out.println();
        }
        scn.close();
    }

    // Function to compute the LCM of two numbers
    private static int lcm(int a, int b) {
        return (a * (b / gcd(a, b)));
    }

    // Function to compute the GCD of two numbers
    private static int gcd(int a, int b) {
        if (b == 0) {
            return a;
        }
        return gcd(b, a % b);
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's analyze the code and identify the potential issues:

### Analysis of the Buggy Code
1. **Initialization**:
   - The array `arr` is initialized with size `n + 1` and all elements are set to 1.
   - This is unnecessary as the array should have size `n`.

2. **Multiplication Logic**:
   - The code iterates over each index `j` from 1 to `n`.
   - For each `j`, it multiplies the elements at indices that are multiples of `j` by `k`.
   - This results in the array elements growing exponentially, which is not the intended behavior.

3. **Output**:
   - The array is printed starting from index 1 to `n`.

### Identifying the Bug
The main issue is in the multiplication logic. The code is multiplying the elements by `k` at each step, which is not the correct way to ensure the array is beautiful. Instead, the elements should be set in a way that ensures the sum of elements at indices divisible by `k` is divisible by `k`.

### Challenging Test Input
To expose this bug, we need a small test case where the multiplication logic will cause the array to grow in an unexpected way, making it clear that the array is not beautiful.

### Test Case
Let's consider a small `n` value to make the output manageable and ensure the bug is evident.

```plaintext
1
4
```

### Explanation
- **Test Case**: `n = 4`
- **Expected Output**: An array of size 4 that is beautiful subject to all `k` from 1 to 4.
- **Buggy Code Output**:
  - The array will be initialized as `[1, 1, 1, 1, 1]`.
  - After the first iteration (`j = 1`), the array will be `[1, 1, 2, 3, 4]`.
  - After the second iteration (`j = 2`), the array will be `[1, 1, 2, 6, 12]`.
  - After the third iteration (`j = 3`), the array will be `[1, 1, 2, 6, 36]`.
  - After the fourth iteration (`j = 4`), the array will be `[1, 1, 2, 6, 144]`.

The final array `[1, 1, 2, 6, 144]` is not beautiful because the sum of elements at indices divisible by `k` for some `k` values will not be divisible by `k`.

### Conclusion
The test input `1\n4` will expose the bug in the provided code by showing that the array generated does not satisfy the conditions for being beautiful.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
#include<bits/stdc++.h>
#define hossam ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
#define For(m,b) for(ll m = 0;m<b;m++)
#define DFor(b) for(ll i=0 , int j=0;j<b;)
#define Sfor(m,l,n) for(ll m=l;m<=n;m++)
#define decFor(m,l,n) for(ll m=l;m>=n;m--)
#define AFor(m,a)   for(auto &m:a)
#define VBCA(a) char a[100000]
#define VBLLA(a) ll a[200005]
#define ldoubleVector(a,size) vector<ldb>a(size);p For(i,size){cin>>a[i];}
#define llongVector(a,size) vector<char>a(size); AFor(i,a){cin>>i;};
#define vectorPrint(a) AFor(i,a){cout<<i<<" " ;}cout<<el;
#define SortedLlongVector(a,size) llongVector(a,n) sort(a.begin(),a.end());
#define ll long long
#define db double
#define ldb long double
#define all(a) (a.begin(),a.end())
#define Rall(a) (a.rbegin(),a.rend()
#define el '\n'
#define cond(i) (i)? puts("YES"): puts("NO");
ll __gcd(ll x, ll y) { return (!x) ? y : __gcd(y % x, x); };
ll lcm(ll x, ll y) { return (x / __gcd(x, y)) * y; }
const int mx = 5e3 + 10;
using namespace std;
ll t = 1, n, m, k;
void SILTARA()
{
	cin >> n;
	For(i, n-1)
	{
		cout << i + 1 << " ";
		if (i == n - 2)
		{
			cout << i+2 <<el;
		}
	}

}
int main()
{
	hossam;
	cin >> t;
	while (t--)
	{
		SILTARA();
	}
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that maximize the failure probability. The buggy code prints a sequence of numbers from 1 to n, which does not necessarily satisfy the conditions of the problem. Specifically, the sum of elements at indices that are multiples of k should be divisible by k.

Let's consider the following steps:

1. **Edge Cases:**
   - **Single Element Array (n = 1):** The array should contain a single element, and that element should be divisible by 1.
   - **Small Arrays (n = 2, 3):** These arrays should be checked to ensure the sums of elements at indices divisible by k are correct.
   - **Larger Arrays (n = 10):** This will test the code with a more complex scenario.

2. **Specific Test Cases:**
   - For **n = 1:** The array should be [1].
   - For **n = 2:** The array should be [2, 2] (sum of elements at indices divisible by 1 and 2 should be divisible by 1 and 2 respectively).
   - For **n = 3:** The array should be [3, 3, 3] (sum of elements at indices divisible by 1, 2, and 3 should be divisible by 1, 2, and 3 respectively).
   - For **n = 10:** The array should be carefully constructed to ensure all conditions are met.

Here is the test input:

```plaintext
4
1
2
3
10
```

### Explanation:
- **n = 1:** The array should be [1]. The sum of the single element is 1, which is divisible by 1.
- **n = 2:** The array should be [2, 2]. The sum of elements at indices divisible by 1 is 4 (2 + 2), which is divisible by 1. The sum of elements at indices divisible by 2 is 2, which is divisible by 2.
- **n = 3:** The array should be [3, 3, 3]. The sum of elements at indices divisible by 1 is 9 (3 + 3 + 3), which is divisible by 1. The sum of elements at indices divisible by 2 is 3, which is divisible by 2. The sum of elements at indices divisible by 3 is 3, which is divisible by 3.
- **n = 10:** The array should be constructed to ensure all conditions are met. A possible array is [10, 10, 10, 10, 10, 10, 10, 10, 10, 10]. The sum of elements at indices divisible by 1 is 100 (10 * 10), which is divisible by 1. The sum of elements at indices divisible by 2 is 50 (10 * 5), which is divisible by 2. The sum of elements at indices divisible by 3 is 30 (10 * 3), which is divisible by 3, and so on.

This test input will help expose the bug in the provided buggy code, which simply prints a sequence of numbers from 1 to n, and does not ensure the required conditions are met.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;

#ifdef local
#include<zhanyi/debug>
#else
#define dbg(...)
#endif

using i64 = long long;

void solve() {
	int n;
	cin >> n;

    for (int i = 1; i <= n; i++) {
    	int res = 1;
    	for (int j = 1; j <= i; j++) {
    		res *= j;
    	}

    	cout << res << " \n"[i == n];
    }
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int t = 1;
    cin >> t;

    while (t--) {
        solve();
    }

    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the buggy code, we need to focus on the core issue in the code. The buggy code generates the array elements by calculating the factorial of the index, which is not guaranteed to satisfy the conditions of the problem. Specifically, the factorial of an index is not necessarily divisible by the index itself, especially for larger indices.

Let's break down the requirements and the bug:

1. **Requirement**: The sum of elements at indices that are multiples of `k` must be divisible by `k` for all `1 <= k <= n`.
2. **Bug**: The code calculates the factorial of the index, which does not ensure that the sum of elements at indices that are multiples of `k` is divisible by `k`.

To expose this bug, we need a test case where the factorial values do not satisfy the divisibility condition. A good approach is to use a small `n` where the factorial values are still manageable but can clearly show the issue.

### Chosen Test Case
Let's choose `n = 4` and analyze the output of the buggy code:

- For `i = 1`, `res = 1! = 1`
- For `i = 2`, `res = 2! = 2`
- For `i = 3`, `res = 3! = 6`
- For `i = 4`, `res = 4! = 24`

The output array would be: `1 2 6 24`

Now, let's check the divisibility conditions:
- For `k = 1`: The sum of all elements is `1 + 2 + 6 + 24 = 33`, which is not divisible by 1.
- For `k = 2`: The sum of elements at indices that are multiples of 2 is `2 + 24 = 26`, which is not divisible by 2.
- For `k = 3`: The sum of elements at indices that are multiples of 3 is `6`, which is divisible by 3.
- For `k = 4`: The sum of elements at indices that are multiples of 4 is `24`, which is divisible by 4.

Clearly, the output array `1 2 6 24` does not satisfy the conditions for `k = 1` and `k = 2`.

### Test Input
```plaintext
1
4
```

This test input will expose the bug in the buggy code by showing that the generated array does not satisfy the divisibility conditions for `k = 1` and `k = 2`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
# author: Anuj (cfkage)
# Byterace 2024, 956d2
import math

def solve():
  n = int(input())

  a = [1] * (n + 1)

  for i in range(1, n + 1):
    for j in range(i, n + 1, i):
      a[j] *= i

  return a[1:]

def main():
  for _ in range(int(input())):
    print(*solve())

if __name__ == '__main__':
  main()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that maximize the failure probability. The buggy code multiplies each element by its index, which can lead to incorrect results for specific input sizes and structures.

### Analysis of the Buggy Code:
1. **Initialization**: The array `a` is initialized with 1s.
2. **Multiplication Loop**: For each index `i` from 1 to `n`, the code multiplies `a[j]` by `i` for every `j` that is a multiple of `i`.

### Potential Issues:
- **Overwriting Values**: The code overwrites values in the array multiple times, which can lead to incorrect sums.
- **Edge Cases**: Small values of `n` can reveal issues more clearly due to fewer overwrites.

### Challenging Test Input:
- **Small `n` Values**: Use small values of `n` to minimize the number of overwrites and make the issue more apparent.
- **Specific `n` Values**: Choose `n` values that have multiple divisors, such as 6 or 8, to ensure the multiplication loop runs multiple times.

### Test Input:
```plaintext
3
3
6
8
```

### Explanation:
- **n = 3**: This is a small array, and the multiplication loop will run a few times, making it easier to see if the values are being overwritten incorrectly.
- **n = 6**: This has multiple divisors (1, 2, 3, 6), which will cause the loop to run multiple times and potentially expose issues with the overwriting logic.
- **n = 8**: This also has multiple divisors (1, 2, 4, 8), providing another opportunity to see if the overwriting logic is correct.

This test input should help expose the bug in the provided buggy code by focusing on small and structured inputs that maximize the failure probability.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import math
for _ in range(int(input())):
    n=int(input())
    x=math.factorial(n)
    print(*([x]*n))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the buggy code, we need to focus on the following aspects:

1. **Edge Cases**: Test with the smallest and largest possible values of `n`.
2. **Divisibility Constraints**: Ensure that the test input can highlight the issue with the sum of elements at indices divisible by `k` not being divisible by `k`.

The buggy code uses the factorial of `n` to fill the array. This approach is incorrect because it does not ensure that the sum of elements at indices divisible by `k` is divisible by `k` for all `k` from 1 to `n`.

### Step-by-Step Analysis:
1. **Smallest `n` (1)**:
   - For `n = 1`, the array should be `[x]` where `x` is any integer.
   - The sum of elements at indices divisible by 1 should be divisible by 1, which is always true.

2. **Largest `n` (100)**:
   - For `n = 100`, the array should have 100 elements.
   - The sum of elements at indices divisible by `k` should be divisible by `k` for all `k` from 1 to 100.
   - Using the factorial of 100 (which is a very large number) will likely fail the divisibility test for smaller `k` values.

3. **Intermediate `n` (e.g., 4, 6, 7)**:
   - For `n = 4`, the array should have 4 elements.
   - For `n = 6`, the array should have 6 elements.
   - For `n = 7`, the array should have 7 elements.
   - These values are chosen to test the divisibility constraints for multiple `k` values.

### Chosen Test Input:
- **Test Case 1**: `n = 1` (smallest `n`)
- **Test Case 2**: `n = 4` (intermediate `n`)
- **Test Case 3**: `n = 6` (intermediate `n` with more constraints)
- **Test Case 4**: `n = 100` (largest `n`)

### Output:
```plaintext
4
1
4
6
100
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import java.awt.*;
import java.io.*;
import java.util.*;
import java.util.List;

public class Main {

    static Map[] dp;
    static void solution() {
        int n = in.nextInt();
        int[] a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = 1;
        }
        for (int i = 1; i <= n; i++) {
            for (int j = i; j <= n; j += i) {
                a[j - 1] *= i;
            }
        }
        for (int i = 0; i < n; i++) {
            out.printf("%d ", a[i]);
        }
        out.println();
    }

    public static void main(String[] args) {
        in = new Scunner();
        int n = in.nextInt();
        while (n-- > 0) {
            solution();
        }
        out.close();
    }

    public static PrintWriter out;
    public static Scunner in;

    public static class Scunner {
        BufferedReader br;
        StringTokenizer st;

        public Scunner() {
            br = new BufferedReader(new InputStreamReader(System.in));
            out = new PrintWriter(new BufferedOutputStream(System.out));
        }

        String next() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }

        String nextLine() {
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }

    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases that can highlight the issues in the algorithm. The code initializes all elements of the array to 1 and then multiplies the elements at positions that are multiples of each integer from 1 to n. This approach can lead to incorrect results, especially when the array size is small or when the values grow too large too quickly.

### Key Points to Consider:
1. **Small Array Size (n = 1 or 2)**: These cases are simple but can reveal if the code handles the smallest possible arrays correctly.
2. **Array Size with Specific Multiples**: For example, n = 4 or n = 6, where the multiples of k can be more complex.
3. **Array Size with Prime n**: This can test the code's ability to handle cases where the multiples are less frequent.

### Test Input:
Let's choose a small array size that can highlight the issue:

```plaintext
3
1
2
4
```

### Explanation:
- **n = 1**: The only element in the array should be divisible by 1. This is a trivial case.
- **n = 2**: The array should have two elements, both of which should be divisible by 1, and the second element should be divisible by 2.
- **n = 4**: This case is more complex. The array should have four elements, and the sums of elements at positions that are multiples of 1, 2, and 4 should be divisible by 1, 2, and 4, respectively.

This test input should help expose any issues in the code's logic for handling small array sizes and the specific requirements for divisibility.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Array Divisibility
time_limit_ms: 1000
memory_limit_mb: 256
Description: An array of integers $$$a_1,a_2,\cdots,a_n$$$ is beautiful subject to an integer $$$k$$$ if it satisfies the following:

- The sum of $$$a_{j}$$$ over all $$$j$$$ such that $$$j$$$ is a multiple of $$$k$$$ and $$$1 \le j \le n $$$, itself, is a multiple of $$$k$$$.
- More formally, if $$$\sum_{k | j} a_{j}$$$ is divisible by $$$k$$$ for all $$$1 \le j \le n$$$ then the array $$$a$$$ is beautiful subject to $$$k$$$. Here, the notation $$${k|j}$$$ means $$$k$$$ divides $$$j$$$, that is, $$$j$$$ is a multiple of $$$k$$$.

It can be shown that an answer always exists.

Input: Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.

The first and only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the array.

Output: For each test case, print the required array as described in the problem statement.

Examples:
input:
3
3
6
7
output:
4 22 18
10 6 15 32 125 54
23 18 27 36 5 66 7

Note: In the second test case, when $$$n = 6$$$, for all integers $$$k$$$ such that $$$1 \le k \le 6$$$, let $$$S$$$ be the set of all indices of the array that are divisible by $$$k$$$.

- When $$$k = 1$$$, $$$S = \{1, 2, 3,4,5,6\}$$$ meaning $$$a_1+a_2+a_3+a_4+a_5+a_6=242$$$ must be divisible by $$$1$$$.
- When $$$k = 2$$$, $$$S = \{2,4,6\}$$$ meaning $$$a_2+a_4+a_6=92$$$ must be divisible by $$$2$$$.
- When $$$k = 3$$$, $$$S = \{3,6\}$$$ meaning $$$a_3+a_6=69$$$ must divisible by $$$3$$$.
- When $$$k = 4$$$, $$$S = \{4\}$$$ meaning $$$a_4=32$$$ must divisible by $$$4$$$.
- When $$$k = 5$$$, $$$S = \{5\}$$$ meaning $$$a_5=125$$$ must divisible by $$$5$$$.
- When $$$k = 6$$$, $$$S = \{6\}$$$ meaning $$$a_6=54$$$ must divisible by $$$6$$$.

Buggy Code:
import java.util.*;

public class Main{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while(t-- != 0){
            int n = sc.nextInt();

            long[] arr = new long[n];
            Arrays.fill(arr,1);
            for(int k=2;k<=n;k++){
                int temp = k;
                for(;temp<=n;temp+=k){
                    arr[temp-1]*=k;
                }
            }

            for(long i : arr){
                System.out.print(i+" ");
            }
            System.out.println();

        }

        sc.close();
    }

}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given code, let's analyze the code and the problem requirements:

### Problem Requirements:
- The array should be beautiful subject to an integer \( k \) if the sum of elements at positions that are multiples of \( k \) is divisible by \( k \).

### Buggy Code Analysis:
- The code initializes an array with all elements set to 1.
- It then iterates over all \( k \) from 2 to \( n \) and multiplies the elements at positions that are multiples of \( k \) by \( k \).

### Potential Bugs:
1. **Initialization and Multiplication**: The code starts with all elements as 1 and multiplies them by \( k \). This can lead to large numbers, which might exceed the expected constraints or cause overflow.
2. **Handling Small Values**: The code might not handle small values of \( n \) correctly, especially when \( n = 1 \) or \( n = 2 \).

### Challenging Test Input:
To expose the bug, we need a test case where the array size is small but the conditions are non-trivial. Let's consider \( n = 4 \):

- For \( n = 4 \):
  - When \( k = 1 \), the sum \( a_1 + a_2 + a_3 + a_4 \) must be divisible by 1.
  - When \( k = 2 \), the sum \( a_2 + a_4 \) must be divisible by 2.
  - When \( k = 3 \), the sum \( a_3 \) must be divisible by 3.
  - When \( k = 4 \), the sum \( a_4 \) must be divisible by 4.

### Test Input:
```plaintext
1
4
```

### Expected Output:
The expected output for \( n = 4 \) should be an array where:
- The sum of all elements is divisible by 1.
- The sum of elements at positions 2 and 4 is divisible by 2.
- The element at position 3 is divisible by 3.
- The element at position 4 is divisible by 4.

A possible correct output could be:
```plaintext
1 2 3 4
```

### Explanation:
- \( 1 + 2 + 3 + 4 = 10 \) (divisible by 1)
- \( 2 + 4 = 6 \) (divisible by 2)
- \( 3 \) (divisible by 3)
- \( 4 \) (divisible by 4)

### Why This Test Input is Challenging:
- The small size of \( n = 4 \) ensures that the output is manageable and easy to verify.
- The conditions are non-trivial and require careful handling of the array elements to ensure all conditions are met.
- The buggy code might not handle the small values correctly, especially when it comes to the multiplication and initialization steps.

This test input should expose the bug in the given code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Gerrymandering
time_limit_ms: 2000
memory_limit_mb: 256
Description: We all steal a little bit. But I have only one hand, while my adversaries have two.

Álvaro Obregón

Álvaro and José are the only candidates running for the presidency of Tepito, a rectangular grid of $2$ rows and $n$ columns, where each cell represents a house. It is guaranteed that $n$ is a multiple of $3$.

Under the voting system of Tepito, the grid will be split into districts, which consist of any $3$ houses that are connected$^{\text{∗}}$. Each house will belong to exactly one district.

Each district will cast a single vote. The district will vote for Álvaro or José respectively if at least $2$ houses in that district select them. Therefore, a total of $\frac{2n}{3}$ votes will be cast.

As Álvaro is the current president, he knows exactly which candidate each house will select. If Álvaro divides the houses into districts optimally, determine the maximum number of votes he can get.

$^{\text{∗}}$A set of cells is connected if there is a path between any $2$ cells that requires moving only up, down, left and right through cells in the set.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains one integer $n$ ($3 \le n \le 10^5$; $n$ is a multiple of $3$) — the number of columns of Tepito.

The following two lines each contain a string of length $n$. The $i$-th line contains the string $s_i$, consisting of the characters $\texttt{A}$ and $\texttt{J}$. If $s_{i,j}=\texttt{A}$, the house in the $i$-th row and $j$-th column will select Álvaro. Otherwise if $s_{i,j}=\texttt{J}$, the house will select José.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.

Output: For each test case, output a single integer — the maximum number of districts Álvaro can win by optimally dividing the houses into districts.

Examples:
input:
4
3
AAA
AJJ
6
JAJAJJ
JJAJAJ
6
AJJJAJ
AJJAAA
9
AJJJJAJAJ
JAAJJJJJA
output:
2
2
3
2

Note: The image below showcases the optimal arrangement of districts Álvaro can use for each test case in the example.

![](CDN_BASE_URL/5f1f1034d2d521a09ea4a9a366636fd8)

Buggy Code:
#include <bits/stdc++.h>
using namespace std;

using i64 = long long;
using u64 = unsigned long long;
using i128 =  __int128;
using ld = long double;
using pii =  pair<int, int>;
using pll =  pair<long long, long long>;
constexpr i64 inf = 0x3f3f3f3f;
constexpr i64 mod = 998244353;

void solve() {

	int n;
	cin >> n;

	vector g(3, vector<int>(n + 1, 0));
	vector dp(3, vector<int>(n + 1, 0));

	char c;

	for(int i = 1; i <= 2; i ++) {
		for(int  j = 1; j <= n; j ++) {
			cin >> c;
			if(c == 'A') {
				g[i][j] = 1;
			} else if(c == 'J') {
				g[i][j] = 0;
			}
		}
	}

	auto check1 = [&](int x, int y) -> bool {
		return (g[x][y] + g[x + 1][y] + g[x][y + 1]) >= 2;
	};

	auto check2 = [&](int x, int y) -> bool {
		return (g[x][y] + g[x - 1][y] + g[x][y + 1]) >= 2;
	};

	auto check3 = [&](int x, int y) -> bool {
		return (g[x][y] + g[x][y - 1] + g[x + 1][y]) >= 2;
	};

	auto check4 = [&](int x, int y) -> bool {
		return (g[x][y] + g[x - 1][y] + g[x][y - 1]) >= 2;
	};

	auto check5 = [&](int x, int y) -> bool {
		return (g[x][y] + g[x][y + 1] + g[x][y + 2]) >= 2;
	};

	for(int i = 3; i <= n; i ++) {
		dp[1][i] = max(dp[1][i], max(dp[1][i - 3], dp[2][i - 3]) + check2(2, i - 2) + check3(1, i));
		dp[2][i] = max(dp[2][i], max(dp[1][i - 3], dp[2][i - 3]) + check1(1, i - 2) + check4(2, i));
		dp[1][i] = dp[2][i] = max({dp[1][i], dp[2][i], max(dp[1][i - 3], dp[2][i - 3]) + check5(1, i - 2) + check5(2, i - 2)});
	}

	for(int i = 6; i <= n; i ++) {
		dp[1][i] = max(dp[1][i], max(dp[1][i - 6], dp[2][i - 6]) + check2(2, i - 5) + check3(1, i) + check5(1, i - 4) + check5(2, i - 3));
		dp[2][i] = max(dp[2][i], max(dp[1][i - 6], dp[2][i - 6]) + check1(1, i - 5) + check4(2, i) + check5(1, i - 3) + check5(2, i - 4));
	}

	// for(int i = 1; i <= 2; i ++) {
	// 	for(int j = 1; j <= n; j ++) {
	// 		cout << dp[i][j] << " \n"[j == n];
	// 	}
	// }

	int ans = 0;

	for(int i = 1; i <= 2; i ++) {
		for(int j = 1; j <= n; j ++) {
			ans = max(ans, dp[i][j]);
		}
	}

	cout << ans << '\n';
}

int main() {

	std::ios::sync_with_stdio(NULL);
    std::cin.tie(nullptr);

    int t = 1;
    std::cin >> t;

    while(t --) {
        solve();
    }

    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given code, we need to focus on edge cases or scenarios where the code might fail. Here are the steps to think through this:

1. **Understand the Bug**: The code is supposed to find the maximum number of districts Álvaro can win by optimally dividing the houses into districts. The code uses dynamic programming to keep track of the maximum number of districts that can be won for each possible state. However, it might have issues with handling certain configurations of houses, especially when the grid is small or has specific patterns.

2. **Consider Edge Cases**:
   - **Small Grid**: Small values of \( n \) (e.g., \( n = 3 \)) can be used to test the base cases.
   - **Uniform Grid**: Grids where all houses select the same candidate (e.g., all houses select Álvaro or all houses select José).
   - **Mixed Grid**: Grids with a mix of Álvaro and José, especially where the pattern is such that it could lead to suboptimal districting.

3. **Generate a Test Input**:
   - **Small Grid with Mixed Selections**: This can help in testing the logic for small \( n \).
   - **Uniform Grid**: This can help in testing the base cases and the logic for uniform selection.
   - **Mixed Grid with Complex Patterns**: This can help in testing the logic for more complex scenarios.

Given the constraints and the need to generate a small-scale test input, let's create a test case that includes a mix of Álvaro and José in a small grid:

```plaintext
3
3
AJA
AJA
6
AJAJAJ
AJAJAJ
9
AJAJAJA
AJAJAJA
```

This test input includes:
- A small grid (3x3) with a mix of Álvaro and José.
- A larger grid (6x2) with a repeating pattern of Álvaro and José.
- An even larger grid (9x2) with a repeating pattern of Álvaro and José.

These cases will help in testing the logic for small grids, repeating patterns, and more complex configurations.

3
3
AJA
AJA
6
AJAJAJ
AJAJAJ
9
AJAJAJA
AJAJAJA


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Gerrymandering
time_limit_ms: 2000
memory_limit_mb: 256
Description: We all steal a little bit. But I have only one hand, while my adversaries have two.

Álvaro Obregón

Álvaro and José are the only candidates running for the presidency of Tepito, a rectangular grid of $2$ rows and $n$ columns, where each cell represents a house. It is guaranteed that $n$ is a multiple of $3$.

Under the voting system of Tepito, the grid will be split into districts, which consist of any $3$ houses that are connected$^{\text{∗}}$. Each house will belong to exactly one district.

Each district will cast a single vote. The district will vote for Álvaro or José respectively if at least $2$ houses in that district select them. Therefore, a total of $\frac{2n}{3}$ votes will be cast.

As Álvaro is the current president, he knows exactly which candidate each house will select. If Álvaro divides the houses into districts optimally, determine the maximum number of votes he can get.

$^{\text{∗}}$A set of cells is connected if there is a path between any $2$ cells that requires moving only up, down, left and right through cells in the set.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains one integer $n$ ($3 \le n \le 10^5$; $n$ is a multiple of $3$) — the number of columns of Tepito.

The following two lines each contain a string of length $n$. The $i$-th line contains the string $s_i$, consisting of the characters $\texttt{A}$ and $\texttt{J}$. If $s_{i,j}=\texttt{A}$, the house in the $i$-th row and $j$-th column will select Álvaro. Otherwise if $s_{i,j}=\texttt{J}$, the house will select José.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.

Output: For each test case, output a single integer — the maximum number of districts Álvaro can win by optimally dividing the houses into districts.

Examples:
input:
4
3
AAA
AJJ
6
JAJAJJ
JJAJAJ
6
AJJJAJ
AJJAAA
9
AJJJJAJAJ
JAAJJJJJA
output:
2
2
3
2

Note: The image below showcases the optimal arrangement of districts Álvaro can use for each test case in the example.

![](CDN_BASE_URL/5f1f1034d2d521a09ea4a9a366636fd8)

Buggy Code:
#include <bits/stdc++.h>
using ll = long long;
using namespace std;
const int inf = 1e9 + 10;

vector <vector <int>> a;

int col(int l, int r, int c, int d) {
    int cl = 0;
    for (int x = l; x <= r; x++) {
        cl += a[0][x];
    }
    for (int x = c; x <= d; x++) {
        cl += a[1][x];
    }
    return (cl > 1 ? 1 : 0);
}
void solve() {
    int n; cin >> n;
    a.assign(2, vector <int>(n, 0));
    for (int j = 0; j < 2; j++) {
        for (int i = 0; i < n; i++) {
            char x; cin >> x;
            if (x == 'A') {
                a[j][i] = 1;
            }
        }
    }
    vector <vector <int>> dp(n + 1, vector <int>(3, -inf));
    dp[0][0] = 0;
    for (int i = 0; i < n; i++) {
        for (int last_mask = 0; last_mask < 3; last_mask++) {
            int ind = i - 1;
            if (dp[i][last_mask] != -inf) {
                if (last_mask == 0) {
                    if (i + 3 <= n) {
                        dp[i + 3][0] = max(dp[i + 3][0], dp[i][last_mask] + col(i, i + 2, i, i + 2));
                    }
                    if (i + 1 < n) {
                        dp[i + 1][2] = max(dp[i + 1][2], dp[i][last_mask] + col(i, i + 1, i, i));
                        dp[i + 1][1] = max(dp[i + 1][1], dp[i][last_mask] + col(i, i, i, i + 1));
                    }
                } else if (last_mask == 1) {
                    if (i + 3 < n) {
                        dp[i + 3][1] = max(dp[i + 3][1], dp[i][last_mask] + col(i, i + 2, i + 1, i + 3));
                    }
                    if (i + 2 <= n) {
                        dp[i + 2][0] = max(dp[i + 2][0], dp[i][last_mask] + col(i, i + 1, i + 1, i + 1));
                    }
                } else {
                    if (i + 3 < n) {
                        dp[i + 3][2] = max(dp[i + 3][2], dp[i][last_mask] + col(i + 1, i + 3, i, i + 2));
                    }
                    if (i + 2 <= n) {
                        dp[i + 2][0] = max(dp[i + 2][0], dp[i][last_mask] + col(i + 1, i + 1, i, i + 1));
                    }
                }
            }
        }
    }
    cout << dp[n][0] << endl;
}

signed main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    int test = 1;
    cin >> test;
    for (int j = 0; j < test; j++) {
        solve();
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider edge cases and scenarios where the code might fail. Here are the steps to create such a test input:

1. **Identify Potential Bugs**:
   - The code uses dynamic programming to solve the problem, and it relies on the `col` function to count the number of Álvaro votes in a district.
   - The `col` function might have issues with edge cases or incorrect boundary conditions.
   - The DP transitions might have incorrect logic for handling different masks and indices.

2. **Consider Edge Cases**:
   - Small grids where the number of columns is exactly 3.
   - Grids where the number of columns is a multiple of 3 but has a specific pattern that might cause the `col` function to fail.
   - Grids where the distribution of 'A' and 'J' is such that the optimal solution is not straightforward.

3. **Create a Test Case**:
   - Let's create a test case with a small grid that has a specific pattern to challenge the `col` function and DP transitions.

Here is a small-scale test input that might expose the bug:

```plaintext
1
6
AAAJJJ
JJJAAA
```

**Explanation**:
- This test case has a grid with 6 columns.
- The first row has the first three columns as 'A' and the last three columns as 'J'.
- The second row has the first three columns as 'J' and the last three columns as 'A'.
- This pattern ensures that the optimal solution is not straightforward and requires careful handling of the `col` function and DP transitions.

This test case should help in identifying any issues in the provided code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Gerrymandering
time_limit_ms: 2000
memory_limit_mb: 256
Description: We all steal a little bit. But I have only one hand, while my adversaries have two.

Álvaro Obregón

Álvaro and José are the only candidates running for the presidency of Tepito, a rectangular grid of $2$ rows and $n$ columns, where each cell represents a house. It is guaranteed that $n$ is a multiple of $3$.

Under the voting system of Tepito, the grid will be split into districts, which consist of any $3$ houses that are connected$^{\text{∗}}$. Each house will belong to exactly one district.

Each district will cast a single vote. The district will vote for Álvaro or José respectively if at least $2$ houses in that district select them. Therefore, a total of $\frac{2n}{3}$ votes will be cast.

As Álvaro is the current president, he knows exactly which candidate each house will select. If Álvaro divides the houses into districts optimally, determine the maximum number of votes he can get.

$^{\text{∗}}$A set of cells is connected if there is a path between any $2$ cells that requires moving only up, down, left and right through cells in the set.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains one integer $n$ ($3 \le n \le 10^5$; $n$ is a multiple of $3$) — the number of columns of Tepito.

The following two lines each contain a string of length $n$. The $i$-th line contains the string $s_i$, consisting of the characters $\texttt{A}$ and $\texttt{J}$. If $s_{i,j}=\texttt{A}$, the house in the $i$-th row and $j$-th column will select Álvaro. Otherwise if $s_{i,j}=\texttt{J}$, the house will select José.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.

Output: For each test case, output a single integer — the maximum number of districts Álvaro can win by optimally dividing the houses into districts.

Examples:
input:
4
3
AAA
AJJ
6
JAJAJJ
JJAJAJ
6
AJJJAJ
AJJAAA
9
AJJJJAJAJ
JAAJJJJJA
output:
2
2
3
2

Note: The image below showcases the optimal arrangement of districts Álvaro can use for each test case in the example.

![](CDN_BASE_URL/5f1f1034d2d521a09ea4a9a366636fd8)

Buggy Code:
#include <iostream>

using namespace std;

typedef long long ll;

int main(){
    ll t;
    cin >> t;
    while(t--){
        ll n, blocks, answer, countA_row0, countA_row1, count_A1, countA_tile1_a, countA_tile2_a, count_A2a, countA_tile1_b, countA_tile2_b, count_A2b, count_A_block;
        cin >> n;
        string s0, s1;
        cin >> s0 >> s1;
        blocks = n / 3;
        answer = 0;
        for (ll b=0; b<blocks; b++){
            ll i = b * 3;
            countA_row0 = 0;
            if (s0[i] == 'A') {
                countA_row0++;
            }
            if (s0[i + 1] == 'A') {
                countA_row0++;
            }
            if (s0[i + 2] == 'A') {
                countA_row0++;
            }
            countA_row1 = 0;
            if (s1[i] == 'A') {
                countA_row1++;
            }
            if (s1[i + 1] == 'A') {
                countA_row1++;
            }
            if (s1[i + 2] == 'A') {
                countA_row1++;
            }
            count_A1 = 0;
            if (countA_row0 >= 2) {
                count_A1 +=1;
            }
            if (countA_row1 >=2) {
                count_A1 +=1;
            }
            countA_tile1_a = 0;
            if (s0[i] == 'A') {
                countA_tile1_a++;
            }
            if (s0[i + 1] == 'A') {
                countA_tile1_a++;
            }
            if (s1[i] == 'A') {
                countA_tile1_a++;
            }
            bool A_tile1_a = (countA_tile1_a >=2) ? 1 : 0;
            countA_tile2_a = 0;
            if (s1[i + 1] == 'A') {
                countA_tile2_a++;
            }
            if(s0[i + 2] == 'A') {
                countA_tile2_a++;
            }
            if(s1[i + 2] == 'A') {
                countA_tile2_a++;
            }
            bool A_tile2_a = (countA_tile2_a >=2) ? 1 : 0;
            count_A2a = A_tile1_a + A_tile2_a;
            countA_tile1_b = 0;
            if (s1[i] == 'A') {
                countA_tile1_b++;
            }
            if (s1[i + 1] == 'A') {
                countA_tile1_b++;
            }
            if (s0[i] == 'A') {
                countA_tile1_b++;
            }
            bool A_tile1_b = (countA_tile1_b >=2) ? 1 : 0;
            countA_tile2_b = 0;
            if (s0[i + 1] == 'A') {
                countA_tile2_b++;
            }
            if (s0[i + 2] == 'A') {
                countA_tile2_b++;
            }
            if (s1[i + 2] == 'A') {
                countA_tile2_b++;
            }
            bool A_tile2_b = (countA_tile2_b >=2) ? 1 : 0;
            count_A2b = A_tile1_b + A_tile2_b;
            count_A_block = max({count_A1, count_A2a, count_A2b});
            answer += count_A_block;
        }
        cout << answer << "\n";
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
4
3
AAA
AJJ
6
JAJAJJ
JJAJAJ
6
AJJJAJ
AJJAAA
9
AJJJJAJAJ
JAAJJJJJA
```

4
3
AAA
AJJ
6
JAJAJJ
JJAJAJ
6
AJJJAJ
AJJAAA
9
AJJJJAJAJ
JAAJJJJJA