CF1792E Divisors and Table

You are given an $ n \times n $ multiplication table and a positive integer $ m = m_1 \cdot m_2 $ . A $ n \times n $ multiplication table is a table with $ n $ rows and $ n $ columns numbered from $ 1 $ to $ n $ , where $ a_{i, j} = i \cdot j $ . For each divisor $ d $ of $ m $ , check: does $ d $ occur in the table at least once, and if it does, what is the minimum row that contains $ d $ .

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 $ .

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](https://en.wikipedia.org/wiki/Bitwise_operation#XOR).

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: 123112322463369 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: 1234561123456224681012336912151844812162024551015202530661218243036 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 $ .