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 proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
#[allow(unused_imports)]
use std::f64::consts::*;

#[allow(unused)]
const INF: usize = std::usize::MAX / 4;
#[allow(unused)]
const M: usize = 1000000007;

fn factors(n: usize) -> Vec<usize> {
    let mut result = vec![];
    for d in 1.. {
        if d * d > n {
            break;
        }
        if n % d == 0 {
            result.push(d);
            result.push(n / d);
        }
    }
    result.sort();
    result.dedup();
    result
}

fn main() {
    input! {
        n: usize,
    }
    let mut result = HashSet::new();
    for &d in &factors(n - 1) {
        if d > 1 {
            result.insert(d);
        }
    }
    for &d in &factors(n) {
        if d == 1 {
            continue;
        }
        let mut m = n;
        while m % d == 0 {
            m /= d;
        }
        if m % d == 1 {
            result.insert(d);
        }
    }

    println!("{}", result.len());
}