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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
public static class P
{
    public static void Main()
    {
        var n = long.Parse(Console.ReadLine());
        HashSet<long> res = new HashSet<long>();
        {
            res.Add(n - 1);
            long i;
            for (i = 2; i * i < n - 1; i++)
            {
                if ((n - 1) % i != 0) continue;
                res.Add(i);
                res.Add((n - 1) / i);
            }
            if (i * i == n - 1) res.Add(i);
        }
        {
            res.Add(n);
            
            long i;
            for (i = 2; i * i < n; i++)
            {
                if (n % i != 0) continue;
                if (isValid(i, n)) res.Add(i);
                if (isValid(n / i, n)) res.Add(n / i);
            }
            if (i * i == n)
                if (isValid(i, n)) res.Add(i);
        }
        Console.WriteLine(res.Count);
    }
    static bool isValid(long i, long n )
    {
        var tmp = n;
        while (tmp % i == 0) tmp /= i;
        return (tmp - 1) % i == 0;
    }
}