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 Rust (1.42.0) (judge verdict: Accepted)

use text_io::*;

fn main(){
  let n:usize = read!();
  
  let mut acc = 0;
  for i in 1..=(n-1){
    if i*i>(n-1){break;}
    else if i*i==(n-1){acc+=1;}
    else if (n-1)%i==0{acc+=2;}
  }
  
  acc-=1;
  
  let m=n;
  let mut ac=0;
  
  for i in 1..m{
    if i*i>n{break;}
    else if i*i==n{if judge(n,i){ac+=1;}}
    else if m%i==0{
      let j = m/i;
      if judge(n,i){ac+=1;}
      if judge(n,j){ac+=1;}
    }
  }
    
    println!("{}",acc+ac);
}

fn judge(n:usize,i:usize)->bool{
  if i==1{false}
  else if n%i==0{judge(n/i,i)}
  else {n%i==1}
}