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.Generic;
using System.ComponentModel.Design.Serialization;
using System.Linq;

namespace ABC161F
{
    class Program
    {
        static void Main(string[] args)
        {
            long N = long.Parse(Console.ReadLine());

            int root = (int)Math.Pow(N, 0.5);

            //素因数分解
            long N1 = N;
            long N2 = N - 1;
            List<long> divN1 = new List<long>();
            List<long> divN1key = new List<long>();
            List<long> divN2 = new List<long>();
            List<long> divN2key = new List<long>();
            for (int i = 2; i <= root; i++)
            {
                while (N1 % i == 0)
                {
                    divN1.Add(i);
                    if (!divN1key.Contains(i)) divN1key.Add(i);
                    N1 /= i;
                }
                while (N2 % i == 0)
                {
                    divN2.Add(i);
                    if (!divN2key.Contains(i)) divN2key.Add(i);
                    N2 /= i;
                }
            }
            divN1.Add((int)N1);
            divN1key.Add((int)N1);
            divN2.Add((int)N2);
            divN2key.Add((int)N2);

            int ans = 1;
            for (int i = 0; i < divN2key.Count; i++)
            {
                ans *= (divN2.Count(x => x == divN2key[i]) + 1);
            }

            for (int i = 0; i < divN1key.Count; i++)
            {
                long div = N;
                while (div % divN1key[i] == 0) div /= divN1key[i];

                if (div % divN1key[i] == 1) ans++;
            }

            Console.WriteLine(ans);
            Console.ReadKey();
        }

        static int[] ReadInt()
        {
            int[] ret = Console.ReadLine().Split().Select(int.Parse).ToArray();
            return ret;
        }
        static long[] ReadLong()
        {
            long[] ret = Console.ReadLine().Split().Select(long.Parse).ToArray();
            return ret;
        }
    }
}