cf 983A Finite or not?

最新推荐文章于 2024-11-02 00:41:06 发布

max_kibble

最新推荐文章于 2024-11-02 00:41:06 发布

阅读量450

点赞数

CC 4.0 BY-SA版权

分类专栏： cf 数论文章标签：数论

本文链接：https://blog.youkuaiyun.com/max_kibble/article/details/80339811

cf 同时被 2 个专栏收录

32 篇文章

订阅专栏

数论

2 篇文章

订阅专栏

一原题

You are given several queries. Each query consists of three integers $p$ , $q$ and $b$ . You need to answer whether the result of $p / q$ in notation with base $b$ is a finite fraction.

A fraction in notation with base $b$ is finite if it contains finite number of numerals after the decimal point. It is also possible that a fraction has zero numerals after the decimal point.

Input

The first line contains a single integer $n$ ( $1 \leq n \leq 10^{5}$ ) — the number of queries.

Next $n$ lines contain queries, one per line. Each line contains three integers $p$ , $q$ , and $b$ ( $0 \leq p \leq 10^{18}$ , $1 \leq q \leq 10^{18}$ , $2 \leq b \leq 10^{18}$ ). All numbers are given in notation with base $10$ .

Output

For each question, in a separate line, print Finite if the fraction is finite and Infinite otherwise.

Examples
inputCopy
2
6 12 10
4 3 10
outputCopy
Finite
Infinite
inputCopy
4
1 1 2
9 36 2
4 12 3
3 5 4
outputCopy
Finite
Finite
Finite
Infinite

Note

$\frac{6}{12} = \frac{1}{2} = 0, 5_{10}$

$\frac{4}{3} = 1, (3)_{10}$

$\frac{9}{36} = \frac{1}{4} = 0, 01_{2}$

$\frac{4}{12} = \frac{1}{3} = 0, 1_{3}$

二分析

题意是说给你三个数p,q,b.问你分数p/q在b进制下是否是有穷小数。首先把q除去p,q的最大公约数得到新的q.p/q是有穷小数等价于新的q的素因子都是b的素因子。因为p,q,b最大都是max=10^18，直接分解素因子需要sqrt(max)次操作，会超时。我在比赛的时候想到的办法是求q和b的最大公约数t，只要分解q/t和t中较小的一个就可以了，复杂度就变成了两次开根。更多人选择的解法见下面代码，话说c++17已经自带了std::gcd函数，在头文件<numeric>里。

三代码

#include <cstdio>
#include <numeric>

typedef long long LL;

using std::gcd;

int n;
LL p, q, b;

int main() {
	scanf("%d", &n);
	while (n--) {
		scanf("%lld%lld%lld", &p, &q, &b);
		q /= gcd(p, q);
		while (true) {
			LL g = gcd(q, b);
			if (g == 1) break;
			while (q % g == 0) q /= g;
		}
		if (q == 1) puts("Finite");
		else puts("Infinite");
	}
	return 0;
}