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 (1.15.1) (judge verdict: Accepted)

use std::io::*;
use std::str::FromStr;
use std::collections::HashMap;
use std::collections::HashSet;

fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char) 
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

fn divisors(n: usize) -> Vec<usize> {
    let mut divisors = Vec::new();
    for i in 1..(f64::sqrt(n as f64) + 1e-9) as usize + 1 {
        if n % i == 0 {
            divisors.push(i);

            if i != n / i {
                divisors.push(n / i);
            }
        }
    }
    divisors.sort();
    divisors
}

fn check(n: usize, k: usize) -> bool {
    let mut n = n;
    while n > 1 {
        // println!("{} {}", n, k);
        if n % k == 0 {
            n = n / k;
        } else {
            n = n % k;
            break;
        }
    }
    return n == 1;
}

fn main() {
    let n: usize = read();
    let divisors1 = divisors(n);
    let divisors2 = divisors(n-1);
    // println!("{:?}", divisors1);
    // println!("{:?}", divisors2);

    let mut ans_vec = Vec::new();
    for divs in vec![divisors1, divisors2] {
        for k in divs {
            if k == 1 {
                continue;
            }
            if check(n, k) {
                ans_vec.push(k);
            }
        }
    }

    let uniq: HashSet<usize> = ans_vec.into_iter().collect();
    println!("{}", uniq.len());
}