Divisors and Table

Description:

Input:

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10$$$) — the number of test cases.

The first and only line of each test case contains three integers $$$n$$$, $$$m_1$$$ and $$$m_2$$$ ($$$1 \le n \le 10^9$$$; $$$1 \le m_1, m_2 \le 10^9$$$) — the size of the multiplication table and the integer $$$m$$$ represented as $$$m_1 \cdot m_2$$$.

Output:

For each test case, let $$$d_1, d_2, \dots, d_k$$$ be all divisors of $$$m$$$ sorted in the increasing order. And let $$$a_1, a_2, \dots, a_k$$$ be an array of answers, where $$$a_i$$$ is equal to the minimum row index where divisor $$$d_i$$$ occurs, or $$$0$$$, if there is no such row.

Since array $$$a$$$ may be large, first, print an integer $$$s$$$ — the number of divisors of $$$m$$$ that are present in the $$$n \times n$$$ table. Next, print a single value $$$X = a_1 \oplus a_2 \oplus \dots \oplus a_k$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

Sample Input:

3
3 72 1
10 10 15
6 1 210

Sample Output:

6 2
10 0
8 5

Note:

In the first test case, $$$m = 72 \cdot 1 = 72$$$ and has $$$12$$$ divisors $$$[1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]$$$. The $$$3 \times 3$$$ multiplication table looks like that:

	1	2	3
1	1	2	3
2	2	4	6
3	3	6	9

For each divisor of $$$m$$$ that is present in the table, the position with minimum row index is marked. So the array of answers $$$a$$$ is equal to $$$[1, 1, 1, 2, 2, 0, 3, 0, 0, 0, 0, 0]$$$. There are only $$$6$$$ non-zero values, and xor of $$$a$$$ is equal to $$$2$$$.

In the second test case, $$$m = 10 \cdot 15 = 150$$$ and has $$$12$$$ divisors $$$[1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150]$$$. All divisors except $$$75$$$ and $$$150$$$ are present in the $$$10 \times 10$$$ table. Array $$$a$$$ $$$=$$$ $$$[1, 1, 1, 1, 1, 1, 3, 5, 3, 5, 0, 0]$$$. There are $$$10$$$ non-zero values, and xor of $$$a$$$ is equal to $$$0$$$.

In the third test case, $$$m = 1 \cdot 210 = 210$$$ and has $$$16$$$ divisors $$$[1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210]$$$. The $$$6 \times 6$$$ table with marked divisors is shown below:

	1	2	3	4	5	6
1	1	2	3	4	5	6
2	2	4	6	8	10	12
3	3	6	9	12	15	18
4	4	8	12	16	20	24
5	5	10	15	20	25	30
6	6	12	18	24	30	36

Array $$$a$$$ $$$=$$$ $$$[1, 1, 1, 1, 1, 0, 2, 0, 3, 0, 5, 0, 0, 0, 0, 0]$$$. There are $$$8$$$ non-zero values, and xor of $$$a$$$ is equal to $$$5$$$.