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 <iostream>
#include <iomanip>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <map>
#include <set>
#include <queue>
#include <stack>
#include <vector>
#include <algorithm>
#include <numeric>
#include <random>
#include <bitset>
#include <chrono>

using namespace std;

#define pb push_back
#define fast_cin() ios_base::sync_with_stdio(false); cin.tie(NULL)
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());

typedef long long ll;
typedef unsigned long long ull;
typedef long double ld;
typedef pair <int, int> pii;
typedef pair <ll, ll> pll;

#ifndef ONLINE_JUDGE
// #include "debug.h"
#else
// #define debug(...) 42
#endif

const int M = 2e5+5;
const int MD=1e9+7;


ll div(ll x)
{
	ll cnt=1;
	for(ll i=2;i*i<=x;i++){
		if(x%i==0){
			if(i==x/i) cnt++;
			else cnt+=2;
		}
	}
	return cnt;
}
int main() {
	fast_cin();
	ll n,ans=1,tmp;
	cin>>n;
	ans+=div(n-1);
	tmp=n;
	for(ll i=2;i*i<=n;i++){
		if(n%i) continue;
		ll mul=1;
		while(n%i==0)
		{
			n/=i;
			mul*=i;
		}
		ll xx=tmp/mul;
		if(xx!=1 && xx%n==1) ans++;
	}
	if(n>1){
		ll xx=tmp/n;
		if(xx!=1 && xx%n==1) ans++;
	}
	cout<<ans;
	return 0;
}