数论之路慢慢之GCD性质

题目链接

Strange Optimization

Bobo is facing a strange optimization problem. Given
n,m,
he is going to find a real number
α
such that f(12+α)
is maximized, where
f(t)=mini,j∈Z|in−jm+t|.
Help him!

Note: It can be proved that the result is always rational.

Input

The input contains zero or more test cases and is terminated by end-of-file.

Each test case contains two integers
n,m.

1≤n,m≤109
The number of tests cases does not exceed
104.

Output

For each case, ou
cbde
tput a fraction
p/q
which denotes the result.

Sample Input

1 1
1 2

Sample Output

1/2
1/4

题解：



先把i/n-j/m通分，得出（i*m-j*n）/n*m   然后显然可以提出gcd（n，m）因子

就相当于是k*gcd（n，m）/n*m，k为整数，就是个等差数列，然后就相当于去找符合条件的首项，那么由于是绝对值，，不用考虑最小边界的问题，那么容易得出首项为二分之一公差答案

这个题不要忘了化简，，还有必须I64d才能过题

#include<stdio.h>
#include<stdlib.h>
#include<string.h>
#include<math.h>
#include<iostream>
#include<algorithm>
#include<stack>
#include<queue>
#include<vector>
#include<set>
#include<map>
#include<string>
using namespace std;
typedef long long ll;
typedef pair<int,int>P;
const int INF=0x3f3f3f3f;
ll gcd(ll a,ll b)
{
if(b==0)return a;
return gcd(b,a%b);
}
int main()
{
ll n,m;
while(scanf("%I64d%I64d",&n,&m)!=EOF)
{
ll p=gcd(n,m);
ll q=2*(n*m);
ll cc=gcd(p,q);
printf("%I64d/%I64d\n",p/cc,q/cc);
}
return 0;
}