C. The Big Race
time limit per test1 second
memory limit per test256 megabytes
inputstandard input
outputstandard output
Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.
Willman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner.
While watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes).
Note that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.
Since the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?
Input
The first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·1018) — the maximum possible length of the racetrack, the length of Willman’s steps and the length of Bolt’s steps respectively.
Output
Print the answer to the problem as an irreducible fraction . Follow the format of the samples output.
The fraction (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.
Sample test(s)
input
10 3 2
output
3/10
input
7 1 2
output
3/7
Note
In the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.
思路：
比賽爲不公平的情況下：
1、兩者都跳進坑裏，即兩者公倍數
2、兩者都不跳進坑裏，即1～min(w, b)-1
3、當長度t小於公倍數時， sum = min(w, b)*(n/lcm) + min(n%lcm, min(w, b)-1);
t大於公倍數時， sum = min(min(w, b) - 1, n);
注意
1、本題在求公倍數時有可能出現溢出情況，需先w／gcd（w, b）*b求出公倍數
2、由於w，b可能爲兩個大數，則其公倍數肯定溢出，而長度t一定小於其公倍數。所以可以用n/w*gcd（w, b）/b判斷是否t小於其公倍數。
#include <cstdio>
#include <algorithm>
#include <iostream>
#include <cctype>
using namespace std;
typedef long long LL;
LL n, w, b, m, sum, d;
LL gcd(LL a, LL b) {
return b == 0? a: gcd(b, a%b);
}
int main() {
cin >> n >> w >> b;
if (n/w*gcd(w, b)/b > 0) {
LL lcm = w/gcd(w, b)*b;
sum = min(w, b)*(n/lcm) + min(n%lcm, min(w, b)-1);
} else {
sum = min(min(w, b) - 1, n);
}
d = gcd(sum, n);
printf("%lld/%lld\n", sum/d, n/d);
return 0;
}