codeforces #483B# Friends and Presents（二分+math）_friends and presents 二分法-优快云博客

解决一个关于礼物分配的问题，通过数学和集合的方法找到最小数值v，确保礼物可以公平地分配给两个朋友，同时考虑到他们不喜欢的特定类型礼物。

摘要生成于 C知道，由 DeepSeek-R1 满血版支持，前往体验 >

You have two friends. You want to present each of them several positive integers. You want to present cnt1 numbers to the first friend and cnt2 numbers to the second friend. Moreover, you want all presented numbers to be distinct, that also means that no number should be presented to both friends.

In addition, the first friend does not like the numbers that are divisible without remainder by prime number x. The second one does not like the numbers that are divisible without remainder by prime number y. Of course, you're not going to present your friends numbers they don't like.

Your task is to find such minimum number v, that you can form presents using numbers from a set 1, 2, ..., v. Of course you may choose not to present some numbers at all.

A positive integer number greater than 1 is called prime if it has no positive divisors other than 1 and itself.

Input

The only line contains four positive integers cnt1, cnt2, x, y (1 ≤ cnt1, cnt2 < 109; cnt1 + cnt2 ≤ 109; 2 ≤ x < y ≤ 3·104) — the numbers that are described in the statement. It is guaranteed that numbers x, y are prime.

Output

Print a single integer — the answer to the problem.

Sample test(s)

input
3 1 2 3

output
5

input
1 3 2 3

output
4

Note

In the first sample you give the set of numbers {1, 3, 5} to the first friend and the set of numbers {2} to the second friend. Note that if you give set {1, 3, 5} to the first friend, then we cannot give any of the numbers 1, 3, 5 to the second friend.

In the second sample you give the set of numbers {3} to the first friend, and the set of numbers {1, 2, 4} to the second friend. Thus, the answer to the problem is 4.

解题思路：这道题其实是一道和集合有关的数学题，如果你直接枚举肯定是超时。

给定一个素数p和一个范围[1，n]，那么能整除的个数就是n / p。

现在相当于用二分法求一个最小的n使得， [1,n]内的数中非x倍数的数给a做礼物的个数 >= cnt1，非y倍数的数给b做为礼物的个数 >= cnt2。并且作为礼物送给a后就不能送给b了。主要是有一部分礼物既可以给a做礼物也可以给b做礼物，要搞清楚交集。同时这题的用int会超出范围，导致超时，用long long。

/********************************/
/*Problem:  codeforces #483B#   */
/*User: 	    shinelin        */
/*Memory: 	    0 KB            */
/*Time: 	    31 ms           */
/*Language: 	GNU C++         */
/********************************/
#include <cstdio>
#include <iostream>
#include <cstdlib>
#include <ctime>
#include <cctype>
#include <cstring>
#include <string>
#include <list>
#include <map>
#include <queue>
#include <deque>
#include <stack>
#include <vector>
#include <set>
#include <algorithm>
#include <cmath>
using namespace std;

#define INF 0x7fffffff
#define LL long long

int cnt1, cnt2, x, y;
LL up = INF, down = 0;
int main()
{
    scanf("%d%d%d%d", &cnt1, &cnt2, &x, &y);
    LL a, b, c, mid;
    a = b = c = 0;
    LL ans = INF;
    while (down <= up)
    {
        mid = (up + down) / 2;
        c = mid - mid / x - mid / y + mid / (x * y);//适合给a,b做礼物
        a = mid / y -  mid / (x * y);               //只能给a做礼物
        b = mid / x -  mid / (x * y);               //只能给b做礼物
        if(a + c >= cnt1 && b + c >= cnt2)
        {
            if(a + b + c == cnt1 + cnt2)
            {
                ans = mid;
                break;
            }
            if(a + b + c > cnt1 + cnt2)
            {
                if(ans > mid)
                    ans = mid;
                up = mid - 1;
            }
            if(a + b + c < cnt1 + cnt2)
            {
                down = mid + 1;
            }
        }
        else
        {
            down = mid + 1;
        }
    }
    printf("%I64d\n", ans);
    return 0;
}