Potion Brewing Class

题意翻译

* 有一个长度为 $n$ 的**正整数**序列 $a$。 * 给定 $n-1$ 条关系：$\frac{a_{u_i}}{a_{v_i}}=\frac{x_i}{y_i}$，保证此时如果已知 $a$ 中任意一个数的值，可以推出剩下所有数。 * 对于所有可能的序列 $a$ 中，求出最小的 $\sum a_i$，对 $998244353$ 取模。 * $T\leq 10^4$，$\sum n\leq 2\times 10^5$，$1\leq u_i,v_i,x_i,y_i\leq n$。

题目描述

Alice's potion making professor gave the following assignment to his students: brew a potion using $ n $ ingredients, such that the proportion of ingredient $ i $ in the final potion is $ r_i > 0 $ (and $ r_1 + r_2 + \cdots + r_n = 1 $ ). He forgot the recipe, and now all he remembers is a set of $ n-1 $ facts of the form, "ingredients $ i $ and $ j $ should have a ratio of $ x $ to $ y $ " (i.e., if $ a_i $ and $ a_j $ are the amounts of ingredient $ i $ and $ j $ in the potion respectively, then it must hold $ a_i/a_j = x/y $ ), where $ x $ and $ y $ are positive integers. However, it is guaranteed that the set of facts he remembers is sufficient to uniquely determine the original values $ r_i $ . He decided that he will allow the students to pass the class as long as they submit a potion which satisfies all of the $ n-1 $ requirements (there may be many such satisfactory potions), and contains a positive integer amount of each ingredient. Find the minimum total amount of ingredients needed to make a potion which passes the class. As the result can be very large, you should print the answer modulo $ 998\,244\,353 $ .

输入输出格式

输入格式

The first line contains a single integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test cases. The first line of each test case contains a single integer $ n $ ( $ 2 \le n \le 2 \cdot 10^5 $ ). Each of the next $ n-1 $ lines contains four integers $ i, j, x, y $ ( $ 1 \le i, j \le n $ , $ i\not=j $ , $ 1\le x, y \le n $ ) — ingredients $ i $ and $ j $ should have a ratio of $ x $ to $ y $ . It is guaranteed that the set of facts is sufficient to uniquely determine the original values $ r_i $ . It is also guaranteed that the sum of $ n $ for all test cases does not exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case, print the minimum total amount of ingredients needed to make a potion which passes the class, modulo $ 998\,244\,353 $ .

输入输出样例

输入样例 #1

3
4
3 2 3 4
1 2 4 3
1 4 2 4
8
5 4 2 3
6 4 5 4
1 3 5 2
6 8 2 1
3 5 3 4
3 2 2 5
6 7 4 3
17
8 7 4 16
9 17 4 5
5 14 13 12
11 1 17 14
6 13 8 9
2 11 3 11
4 17 7 2
17 16 8 6
15 5 1 14
16 7 1 10
12 17 13 10
11 16 7 2
10 11 6 4
13 17 14 6
3 11 15 8
15 6 12 8

输出样例 #1

69
359
573672453

说明

In the first test case, the minimum total amount of ingredients is $ 69 $ . In fact, the amounts of ingredients $ 1, 2, 3, 4 $ of a valid potion are $ 16, 12, 9, 32 $ , respectively. The potion is valid because - Ingredients $ 3 $ and $ 2 $ have a ratio of $ 9 : 12 = 3 : 4 $ ; - Ingredients $ 1 $ and $ 2 $ have a ratio of $ 16 : 12 = 4 : 3 $ ; - Ingredients $ 1 $ and $ 4 $ have a ratio of $ 16 : 32 = 2 : 4 $ . In the second test case, the amounts of ingredients $ 1, 2, 3, 4, 5, 6, 7, 8 $ in the potion that minimizes the total amount of ingredients are $ 60, 60, 24, 48, 32, 60, 45, 30 $ .

题意翻译

* 有一个长度为 $n$ 的**正整数**序列 $a$。 * 给定 $n-1$ 条关系：$\frac{a_{u_i}}{a_{v_i}}=\frac{x_i}{y_i}$，保证此时如果已知 $a$ 中任意一个数的值，可以推出剩下所有数。 * 对于所有可能的序列 $a$ 中，求出最小的 $\sum a_i$，对 $998244353$ 取模。 * $T\leq 10^4$，$\sum n\leq 2\times 10^5$，$1\leq u_i,v_i,x_i,y_i\leq n$。