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>
using namespace std;
typedef long long LL;
#define fi first
#define se second
#define pb push_back
#define forn(i,n) for(int i=0;i<(n);i++)
#define for1(i,n) for(int i=1;i<=n;i++)
#define forr(i,n) for(int i=n;i>=0;i--)
#define all(x) x.begin(), x.end()
const int MAXN = 1e5 +5;
using u64 = LL;
using u128 = LL;

void fio(){
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
}

LL po(LL a, LL n, LL m){
  if(n==0) return 1;
  LL an = po(a,n/2,m);
  an = (an*an)%m;
  if(n&1) return an*a%m;
  return an;
}

bool isprime(int n, int iter=5) {
    if (n < 4)
        return n == 2 || n == 3;

    for (int i = 0; i < iter; i++) {
        int a = 2 + rand() % (n - 3);
        if (po(a, n - 1, n) != 1)
            return false;
    }
    return true;
}


int main(){
  fio();
  LL n; cin>>n;
  LL k = n-1;
  set<LL> s;
  for(LL i = 2;i*i<=k;i++){
    if(k%i==0) s.insert(i), s.insert(k/i);
  }
  s.insert(k);
  for(LL i=3;i*i<n;i++){
    if(n%i==0 && !isprime(n/i)) s.insert(i);
  }
  s.insert(2);
  s.insert(n);
//  for(auto i:s) cout<<i<<" ";
  //  cout<<endl;
  cout<<s.size()<<endl;
  return 0;
}