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;
template <typename A, typename B>
string to_string(pair<A, B> p);

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);

string to_string(const string &s)
{
    return '"' + s + '"';
}

string to_string(const char *s)
{
    return to_string((string)s);
}

string to_string(bool b)
{
    return (b ? "true" : "false");
}

string to_string(vector<bool> v)
{
    bool first = true;
    string res = "{";
    for (int i = 0; i < static_cast<int>(v.size()); i++)
    {
        if (!first)
        {
            res += ", ";
        }
        first = false;
        res += to_string(v[i]);
    }
    res += "}";
    return res;
}

template <size_t N>
string to_string(bitset<N> v)
{
    string res = "";
    for (size_t i = 0; i < N; i++)
    {
        res += static_cast<char>('0' + v[i]);
    }
    return res;
}

template <typename A>
string to_string(A v)
{
    bool first = true;
    string res = "{";
    for (const auto &x : v)
    {
        if (!first)
        {
            res += ", ";
        }
        first = false;
        res += to_string(x);
    }
    res += "}";
    return res;
}

template <typename A, typename B>
string to_string(pair<A, B> p)
{
    return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p)
{
    return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";
}

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p)
{
    return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T)
{
    cerr << " " << to_string(H);
    debug_out(T...);
}

typedef long long ll;
typedef pair<int, int> pii;
#define mp make_pair
#define fst first
#define snd second
#define fr(i, a, b) for(int i=a; i<b; i++)
template<class T = int> using V = vector<T>;
template<class T = int> using VV = V< V<T> >;

int main() {
    cin.tie(nullptr); ios::sync_with_stdio(false);
    ll n; cin >> n;
    set<ll> s;
    for(ll i=2; i*i<=n; i++) {
        ll n2 = n;
        while(n2 % i == 0) {
            n2 /= i;
        }
        if(n2 % i == 1) s.insert(i);
    }
    n--;
    for(ll i=2; i*i<=n; i++) {
        if(n % i == 0) {
            s.insert(i); s.insert(n/i);
        }
    }
    s.insert(n);
    s.insert(n+1);
    cout << s.size() << endl;
    
}