Problem:
Score : 600 points

Problem Statement

Given is a positive integer N.

We will choose an integer K between 2 and N (inclusive), then we will repeat the operation below until N becomes less than K.

Operation: if K divides N, replace N with N/K; otherwise, replace N with N-K.

In how many choices of K will N become 1 in the end?

Constraints

2 \leq N \leq 10^{12}

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of choices of K in which N becomes 1 in the end.

Sample Input 1

6

Sample Output 1

3

There are three choices of K in which N becomes 1 in the end: 2, 5, and 6.

In each of these choices, N will change as follows:

When K=2: 6 \to 3 \to 1

When K=5: 6 \to 1

When K=6: 6 \to 1

Sample Input 2

3141

Sample Output 2

13

Sample Input 3

314159265358

Sample Output 3

9

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<bits/stdc++.h>
#define rep(a,b,c) for(int a=b;a<=c;a++)
#define per(a,b,c) for(int a=b;a>=c;a--)
#define pb push_back
#define mk make_pair
#define pii pair<int,int>
#define fi first
#define se second
#define mem(a,b) memset(a,b,sizeof(a))
#define inf 0x3f3f3f3f
#define ls(x) (x<<1)
#define rs(x) (ls(x)|1)
#define eps 1e-11
using namespace std;
typedef long long ll;
typedef pair<ll,ll> pll;
typedef double db;
const int mod = 998244353;
const int N = 1e6 + 5;
const int M = 1e3 + 5;
ll gcd(ll a,ll b){ return b?gcd(b,a%b):a; }
ll lcm(ll a,ll b){ return a*b/gcd(a,b); }
inline ll qsm(ll x,ll y){ ll res=1; while(y){if(y&1) res=res*x%mod;
x=x*x%mod; y>>=1;} return res; }
ll n,m,k;
int ok(int x,int k){
    m=x;
    while(m%k==0) m/=k;
    return m%k==1;
}


int main(){
    cin>>n;
    m=n-1;
    int ans=1, sz=sqrt(n-1);
    rep(i,2,sz){
        if((n-1)%i==0){
            ++ans;
            if((n-1)/i!=i) ++ans;
        }
    }
    ++ans;
    sz=sqrt(n);
    rep(i,2,sz){
        if(n%i==0&&ok(n,i)) ++ans;//,cout<<"i="<<i<<endl;
    }
    cout<< ans<<endl;
    return 0;
}
/*

*/
//
//                            _ooOoo_
//                           o8888888o
//                           88" . "88
//                           (| -_- |)
//                           O\  =  /O
//                        ____/`---'\____
//                      .'  \|     |//  `.
//                     /  \|||  :  |||//  \
//                    /  _||||| -:- |||||-  \
//                    |   | \  -  /// |   |
//                    | \_|  ''\---/''  |   |
//                    \  .-\__  `-`  ___/-. /
//                  ___`. .'  /--.--\  `. . __
//               ."" '<  `.___\_<|>_/___.'  >'"".
//              | | :  `- \`.;`\ _ /`;.`/ - ` : | |
//              \  \ `-.   \_ __\ /__ _/   .-` /  /
//         ======`-.____`-.___\_____/___.-`____.-'======
//                            `=---='
//        ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
//                     佛祖保佑      永无BUG