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>
typedef long long ll;
typedef long double ld;
using namespace std;

#define REP(i, l, n) for(int i=(l), i##_len=(n); i<i##_len; ++i)
#define ALL(x) (x).begin(),(x).end()
#define pb puah_back

ll gcd(ll a,ll b){return b ? gcd(b,a%b) : a;}
ll lcm(ll a,ll b){return a/gcd(a,b)*b;}

int dx[4] = {-1, 1, 0, 0};
int dy[4] = {0, 0, 1, -1};

vector<ll> enum_div(ll n)//nの約数を列挙
{
	vector<ll> ret;
	for(ll i=1 ; i*i<=n ; ++i)
	{
		if(n%i == 0)
		{
			ret.push_back(i);
			if(i!=1 && i*i!=n)
			{
				ret.push_back(n/i);
			}
		}
	}
    ret.push_back(n);
	return ret;
}

int check(ll n, ll k){
    while(n > k){
        if(n % k == 0){
            n = n / k;
        }else{
            n = n % k;
        }
    }
    if(n == 1){
        return 1;
    }else{
        return 0;
    }
}

int main(){
    ll n; cin >> n;
    ll ans = 1;
    vector<ll> lis1 = enum_div(n);
    REP(i, 1, lis1.size()){
        if(check(n, lis1[i])){
            ans++;
        }
    }
    vector<ll> lis2 = enum_div(n-1);
    REP(i, 1, lis2.size()){
        if(check(n, lis2[i])){
            ans++;
        }
    }
    cout << ans << endl;
}