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)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let n = read::<i64>();
    let nm1 = n - 1;
    let mut i = 1;
    let mut mn_factors = vec![];
    while i * i <= nm1 {
        if nm1 % i == 0 {
            mn_factors.push(i);
            mn_factors.push(nm1 / i);
        }
        i += 1;
    }
    debug!(mn_factors);
    mn_factors.sort();
    mn_factors.dedup();
    let mut ans = mn_factors.len() - 1;

    let mut n_factors = vec![];
    let mut i = 1;
    while i * i <= n {
        if n % i == 0 {
            n_factors.push(i);
            n_factors.push(n / i);
        }
        i += 1;
    }
    n_factors.sort();
    n_factors.dedup();

    debug!(n_factors);
    for factor in n_factors {
        if factor == 1 {
            continue;
        }
        let mut n = n;
        while n % factor == 0 {
            n /= factor;
        }
        if n % factor == 1 {
            ans += 1;
        }
    }

    println!("{}", ans);
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}