HDU-6265 Master of Phi （数论）

2017 杭州CCPC HDU 6265 Master of Phi

You are given an integer n. Please output the answer of ${\sum }_{d|n}\phi (d)\times \frac{n}{d}modulo998244353.$ n is represented in the form of factorization.
φ(n) is Euler’s totient function, and it is defined more formally as the number of integers k in the interval 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1.
For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all co-prime to 9, but the other three numbers in this interval, 3, 6, and 9 are not, because gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only
integer in the interval from 1 to n is 1 itself, and gcd(1, 1) = 1.
And there are several formulas for computing φ(n), for example, Euler’s product formula states like:
$$\varphi (n)=n{\prod }_{p|n}\left(1-\frac{1}{p}\right),$$
where the product is all the distinct prime numbers (p in the formula) dividing n.

Input

The first line contains an integer T (1 ≤ T ≤ 20) representing the number of test cases.
For each test case, the first line contains an integer m (1 ≤ m ≤ 20) is the number of prime factors.
The following m lines each contains two integers $p_i$ and $q_i$ (2 ≤ $p_i$ ≤ 108, 1 ≤ $q_i$ ≤ 108) describing that n contains the factor $p_i^{qi}$, in other words, $n={\prod }_{i=1}^m{p}_i^{q_i}.$ It is guaranteed that all pi are primenumbers and different from each other.

Output

For each test case, print the the answer modulo 998244353 in one line.
Example

standard input	standard output
2 2 2 1 3 1 2 2 2 3 2	15 168

Explanation

For first test case, n = $2^1$ ∗$3^1$ = 6, and the answer is (φ(1)∗n/1+φ(2)∗n/2+φ(3)∗n/3+φ(6)∗n/6) mod 998244353 = (6 + 3 + 4 + 2) mod 998244353 = 15.

题意

首先给你一个数m，接着是m对数$p_i$,$q^{}$