Codeforces Round #382 (Div. 2) -- D. Taxes （数学 -- 哥德巴赫猜想， 唯一分解定理）

大体题意：

你有n元钱，你需要交税，交锐的价格为n 的最大因子（不等于n），为了缩小税钱，你可以拆成多个钱相加的形式，但是不能拆出1来，问你最少交多少钱？

思路：

肯定是全都是素数最好了，那么问题就是把n 拆成多个素数之和的形式！

根据哥德巴赫猜想，任何大于2 的偶数必定写成两个质数相加的形式！

因为我们判断  如果n 本身时素数直接输出1

如果不是素数， 是偶数的话，直接输出2

如果是奇数，两个形式：

1.   2 加上一个 奇素数， 即判断n-2是不是素数  答案是2

2.   一个偶数加上一个奇素数，   答案是3

之所以奇数分成两类  是因为    素数不可能是偶数，奇数不可能由两个素数相加得来！

讨论就好了！

判断素数用唯一分解定理即可判断了！

详细见代码：

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
vector<int>v;
bool is_p(int n){
    int m = sqrt(n) + 1;
    int nn = n;
    v.clear();
    for (int i = 2; i <= m; ++i){
        if (n % i == 0){
            v.push_back(i);
            while(n % i == 0) n/= i;
        }
    }
    if (n != 1) v.push_back(n);
    if ((int)v.size() == 1 && v.back() == nn) return 1;
    return 0;
}
int main(){
    int n;
    scanf("%d",&n);
    if (is_p(n)) return 0 * puts("1");
    if (!(n & 1)) puts("2");
    else {
        if (is_p(n-2)) puts("2");
        else puts("3");
    }
    return 0;
}

D. Taxes

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

Mr. Funt now lives in a country with a very specific tax laws. The total income of mr. Funt during this year is equal to n (n ≥ 2) burles and the amount of tax he has to pay is calculated as the maximum divisor of n (not equal to n, of course). For example, if n = 6 then Funt has to pay 3 burles, while for n = 25 he needs to pay 5 and if n = 2 he pays only 1 burle.

As mr. Funt is a very opportunistic person he wants to cheat a bit. In particular, he wants to split the initial n in several parts n1 + n2 + ... + nk = n (here k is arbitrary, even k = 1 is allowed) and pay the taxes for each part separately. He can't make some part equal to 1 because it will reveal him. So, the condition ni ≥ 2 should hold for all i from 1 to k.

Ostap Bender wonders, how many money Funt has to pay (i.e. minimal) if he chooses and optimal way to split n in parts.

Input

The first line of the input contains a single integer n (2 ≤ n ≤ 2·109) — the total year income of mr. Funt.

Output

Print one integer — minimum possible number of burles that mr. Funt has to pay as a tax.

Examples

input

4

output

2

input

27

output

3