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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.util.*;
import java.io.*;
import static java.lang.Math.*;

public class Main {

    private static final int MOD = (int)1e9 + 7; //998244353;
    private static final long INF = (long)1e15;

    public Main(FastScanner in, PrintWriter out) {
        long N = in.nextLong();
        long res = 0;
        // check all factors of N in O(sqrt(N)log(sqrt(N))) time
        // if is a factor -- find a where a*i^p = N and a % N != 0
        // if a % N == 1, it is a solution
        for (long i = 2; i * i <= N; i++) {
            long x = N;
            while (x % i == 0) {
                x /= i;
            }
            if (N % i == 0 && x % i == 1) {
                res++;
            }
        }
        res += 2; // N, and N - 1 is a solution
        N--;
        // find the number of factors of N - 1 in O(sqt(N)) time
        // if k is a factor of N - 1, it cannot be a factor of N
        // as k is  factor of N - 1, ak + 1 = N
        for (long i = 2; i * i <= N; i++) {
            if (N % i == 0) {
                res++;
                if (i * i != N) res++;
            }
        }
        out.println(res);
    }

    private boolean valid(int n, int k) {
        while (n >= k && n % k == 0) n /= k;
        return n % k == 1;
    }


    private int gcd(int a, int b) {
        while (true) {
            if (b == 0) return a;
            int tmp = a;
            a = b;
            b = tmp % b;
        }
    }

    private long pow(long x, long p) {
        long res = 1;
        while (p > 0) {
            if (p % 2 == 1)
                res = res * x % MOD;
            x = x * x % MOD;
            p /= 2;
        }
        return res;
    }

    public static void main(String[] args) {
        PrintWriter out = new PrintWriter(System.out);
        // Scanner in = new Scanner(
                // new BufferedReader(new InputStreamReader(System.in)));
        FastScanner in = new FastScanner(System.in);

        Main sol = new Main(in, out);

        out.close();
    }
}

// Pair
class Pair<T extends Comparable<? super T>, U extends Comparable<? super U>> implements Comparable<Pair<T, U>> {
    T a;
    U b;
    Pair() { }
    Pair(T a, U b) {
        this.a = a;
        this.b = b;
    }

    @Override
    public String toString() { return "("+a.toString()+", "+b.toString()+")"; }

    @Override
    public int hashCode() { return Objects.hash(a, b); }

    @Override
    public boolean equals(Object obj) {
        if(this == obj) return true;
        if(obj == null) return false;
        if(this.getClass() != obj.getClass()) return false;
        Pair that = (Pair) obj;
        if(this.a.getClass() != that.a.getClass()) return false;
        if(this.b.getClass() != that.b.getClass()) return false;
        if(!this.a.equals(that.a)) return false;
        if(!this.b.equals(that.b)) return false;
        return true;
    }

    @Override
    public int compareTo(Pair<T, U> that) {
        int c = (this.a).compareTo(that.a);
        if(c == 0) c = (this.b).compareTo(that.b);
        return c;
    }
}


class FastScanner{
    private InputStream stream;
    private byte[] buf = new byte[1024];
    private int curChar;
    private int numChars;

    public FastScanner(InputStream stream)
    {
        this.stream = stream;
    }

    int read()
    {
        if (numChars == -1)
            throw new InputMismatchException();
        if (curChar >= numChars){
            curChar = 0;
            try{
                numChars = stream.read(buf);
            } catch (IOException e) {
                throw new InputMismatchException();
            }
            if (numChars <= 0)
                return -1;
        }
        return buf[curChar++];
    }

    boolean isSpaceChar(int c)
    {
        return c==' '||c=='\n'||c=='\r'||c=='\t'||c==-1;
    }

    boolean isEndline(int c)
    {
        return c=='\n'||c=='\r'||c==-1;
    }

    int nextInt()
    {
        return Integer.parseInt(next());
    }

    long nextLong()
    {
        return Long.parseLong(next());
    }

    double nextDouble()
    {
        return Double.parseDouble(next());
    }

    String next(){
        int c = read();
        while (isSpaceChar(c))
            c = read();
        StringBuilder res = new StringBuilder();
        do{
            res.appendCodePoint(c);
            c = read();
        }while(!isSpaceChar(c));
        return res.toString();
    }

    String nextLine(){
        int c = read();
        while (isEndline(c))
            c = read();
        StringBuilder res = new StringBuilder();
        do{
            res.appendCodePoint(c);
            c = read();
        }while(!isEndline(c));
        return res.toString();
    }
}