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)

#![allow(unused_imports)]
#![allow(dead_code)]
use crate::prime_number::{divisors, prime_factors};
use crate::scanner::Scanner;
fn main() {
    let mut scan = Scanner::new();
    let n = scan.read::<i64>();
    let mut cnt = divisors(n - 1).ilen() as i64 - 1;
    for i in divisors(n) {
        if i == 1 {
            continue;
        }
        let mut j = n;
        while j % i == 0 {
            j /= i;
        }
        if j % i == 1 {
            cnt += 1;
        }
    }
    println!("{}", cnt);
}
mod scanner {
    use crate::arraylist::List;
    use std::io::{stdin, BufReader, Bytes, Read, Stdin};
    use std::str::FromStr;
    pub struct Scanner {
        buf: Bytes<BufReader<Stdin>>,
    }
    impl Scanner {
        pub fn new() -> Scanner {
            Scanner {
                buf: BufReader::new(stdin()).bytes(),
            }
        }
        pub fn read_next<T: FromStr>(&mut self) -> Option<T> {
            let token = self
                .buf
                .by_ref()
                .map(|c| c.unwrap() as char)
                .skip_while(|c| c.is_whitespace())
                .take_while(|c| !c.is_whitespace())
                .collect::<String>();
            token.parse::<T>().ok()
        }
        pub fn read<T: FromStr>(&mut self) -> T {
            self.read_next().unwrap()
        }
        pub fn readn<T: FromStr>(&mut self, n: i32) -> List<T> {
            (0..n).map(|_| self.read::<T>()).collect()
        }
        pub fn chars(&mut self) -> List<char> {
            self.read::<String>().chars().collect()
        }
    }
}
mod arraylist {
    use crate::range::IntRangeBounds;
    use std::fmt::Formatter;
    use std::iter::{FromIterator, Sum};
    use std::ops::{Index, IndexMut, RangeBounds};
    use std::slice::Iter;
    #[derive(Clone, PartialEq, Eq)]
    pub struct List<T> {
        pub vec: Vec<T>,
    }
    impl<T> List<T> {
        #[inline]
        pub fn new() -> List<T> {
            List { vec: vec![] }
        }
        #[inline]
        pub fn init(init: T, n: i32) -> List<T>
        where
            T: Clone,
        {
            List {
                vec: vec![init; n as usize],
            }
        }
        #[inline]
        pub fn from_vec(vec: Vec<T>) -> List<T> {
            List { vec }
        }
        #[inline]
        pub fn ilen(&self) -> i32 {
            self.vec.len() as i32
        }
        #[inline]
        pub fn iter(&self) -> Iter<'_, T> {
            self.vec.iter()
        }
        #[inline]
        pub fn push(&mut self, item: T) {
            self.vec.push(item);
        }
        #[inline]
        pub fn sort(&mut self)
        where
            T: Ord,
        {
            self.vec.sort();
        }
        #[inline]
        pub fn reverse(&mut self) {
            self.vec.reverse();
        }
        #[inline]
        pub fn sort_by<F>(&mut self, compare: F)
        where
            F: FnMut(&T, &T) -> std::cmp::Ordering,
        {
            self.vec.sort_by(compare)
        }
        #[inline]
        pub fn sort_by_key<K, F>(&mut self, compare: F)
        where
            F: FnMut(&T) -> K,
            K: Ord,
        {
            self.vec.sort_by_key(compare)
        }
        #[inline]
        pub fn first(&self) -> Option<&T> {
            self.vec.first()
        }
        #[inline]
        pub fn last(&self) -> Option<&T> {
            self.vec.last()
        }
        #[inline]
        pub fn pop(&mut self) -> Option<T> {
            self.vec.pop()
        }
        #[inline]
        pub fn swap(&mut self, i: i32, j: i32) {
            self.vec.swap(i as usize, j as usize);
        }
        #[inline]
        pub fn append(&mut self, mut other: Self) {
            self.vec.append(&mut other.vec);
        }
        #[inline]
        pub fn extend(&mut self, other: impl Iterator<Item = T>) {
            self.vec.extend(other);
        }
        #[inline]
        pub fn mirror(&self) -> std::iter::Cloned<Iter<T>>
        where
            T: Clone,
        {
            self.iter().cloned()
        }
        #[inline]
        pub fn map<B, F>(&self, f: F) -> List<B>
        where
            T: Clone,
            F: FnMut(T) -> B,
        {
            self.mirror().map(f).collect()
        }
        #[inline]
        pub fn filter<P>(&self, predicate: P) -> List<T>
        where
            T: Clone,
            P: FnMut(&T) -> bool,
        {
            self.mirror().filter(predicate).collect()
        }
        #[inline]
        pub fn filter_map<B, F>(&self, f: F) -> List<B>
        where
            T: Clone,
            F: FnMut(T) -> Option<B>,
        {
            self.mirror().filter_map(f).collect()
        }
        #[inline]
        pub fn any<P>(&self, predicate: P) -> bool
        where
            P: FnMut(&T) -> bool,
        {
            self.iter().any(predicate)
        }
        #[inline]
        pub fn all<P>(&self, predicate: P) -> bool
        where
            P: FnMut(&T) -> bool,
        {
            self.iter().all(predicate)
        }
        #[inline]
        pub fn sum<S>(&self) -> S
        where
            T: Clone,
            S: Sum<T>,
        {
            Sum::sum(self.iter().cloned())
        }
        #[inline]
        pub fn enumerate(&self) -> List<(i32, T)>
        where
            T: Clone,
        {
            self.mirror()
                .enumerate()
                .map(|p| (p.0 as i32, p.1))
                .collect()
        }
        #[inline]
        pub fn find<P>(&self, mut predicate: P) -> Option<&T>
        where
            P: FnMut(&T) -> bool,
        {
            self.iter().find(|x| predicate(*x))
        }
        #[inline]
        pub fn index_of<P>(&self, mut predicate: P) -> Option<i32>
        where
            P: FnMut(&T) -> bool,
        {
            self.iter()
                .enumerate()
                .find(|&(_i, x)| predicate(x))
                .map(|p| p.0 as i32)
        }
        #[inline]
        pub fn to<B: FromIterator<T>>(&self) -> B
        where
            T: Clone,
        {
            self.mirror().collect()
        }
        #[inline]
        pub fn min(&self) -> Option<&T>
        where
            T: Ord,
        {
            self.iter().min()
        }
        #[inline]
        pub fn max(&self) -> Option<&T>
        where
            T: Ord,
        {
            self.iter().max()
        }
        #[inline]
        pub fn argmin(&self) -> Option<i32>
        where
            T: Ord,
        {
            let item = self.iter().min()?;
            self.iter()
                .enumerate()
                .find(|p| p.1 == item)
                .map(|p| p.0 as i32)
        }
        #[inline]
        pub fn argmax(&self) -> Option<i32>
        where
            T: Ord,
        {
            let item = self.iter().max()?;
            self.iter()
                .enumerate()
                .find(|p| p.1 == item)
                .map(|p| p.0 as i32)
        }
        #[inline]
        pub fn part<U>(&self, range: U) -> List<T>
        where
            T: Clone,
            U: RangeBounds<i32>,
        {
            List::from_vec(
                self.vec[range.lower_bound(0) as usize..range.upper_bound(self.ilen()) as usize]
                    .to_vec(),
            )
        }
        #[inline]
        pub fn first_exn(&self) -> &T {
            self.first().unwrap()
        }
        #[inline]
        pub fn last_exn(&self) -> &T {
            self.last().unwrap()
        }
        #[inline]
        pub fn pop_exn(&mut self) -> T {
            self.pop().unwrap()
        }
        #[inline]
        pub fn min_exn(&self) -> &T
        where
            T: Ord,
        {
            self.min().unwrap()
        }
        #[inline]
        pub fn max_exn(&self) -> &T
        where
            T: Ord,
        {
            self.max().unwrap()
        }
        #[inline]
        pub fn argmin_exn(&self) -> i32
        where
            T: Ord,
        {
            self.argmin().unwrap()
        }
        #[inline]
        pub fn argmax_exn(&self) -> i32
        where
            T: Ord,
        {
            self.argmax().unwrap()
        }
        #[inline]
        pub fn find_exn<P>(&self, predicate: P) -> &T
        where
            P: FnMut(&T) -> bool,
        {
            self.find(predicate).unwrap()
        }
        #[inline]
        pub fn index_of_exn<P>(&self, predicate: P) -> i32
        where
            P: FnMut(&T) -> bool,
        {
            self.index_of(predicate).unwrap()
        }
    }
    impl<T> Index<i32> for List<T> {
        type Output = T;
        #[inline]
        fn index(&self, index: i32) -> &Self::Output {
            if cfg!(debug_assertions) {
                self.vec.index(index as usize)
            } else {
                unsafe { self.vec.get_unchecked(index as usize) }
            }
        }
    }
    impl<T> IndexMut<i32> for List<T> {
        #[inline]
        fn index_mut(&mut self, index: i32) -> &mut Self::Output {
            if cfg!(debug_assertions) {
                self.vec.index_mut(index as usize)
            } else {
                unsafe { self.vec.get_unchecked_mut(index as usize) }
            }
        }
    }
    impl<T> FromIterator<T> for List<T> {
        fn from_iter<U: IntoIterator<Item = T>>(iter: U) -> Self {
            let mut vec = vec![];
            for i in iter {
                vec.push(i);
            }
            List { vec }
        }
    }
    impl<T> IntoIterator for List<T> {
        type Item = T;
        type IntoIter = std::vec::IntoIter<T>;
        fn into_iter(self) -> std::vec::IntoIter<T> {
            self.vec.into_iter()
        }
    }
    impl<'a, T> IntoIterator for &'a List<T> {
        type Item = &'a T;
        type IntoIter = Iter<'a, T>;
        fn into_iter(self) -> Iter<'a, T> {
            self.vec.iter()
        }
    }
    impl<T: std::fmt::Display> std::fmt::Display for List<T> {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(
                f,
                "{}",
                self.iter()
                    .map(|x| format!("{}", x))
                    .collect::<Vec<_>>()
                    .join(" ")
            )
        }
    }
    impl<T: std::fmt::Debug> std::fmt::Debug for List<T> {
        fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
            write!(
                f,
                "[{}]",
                self.iter()
                    .map(|x| format!("{:?}", x))
                    .collect::<Vec<_>>()
                    .join(", ")
            )
        }
    }
    impl<T> From<Vec<T>> for List<T> {
        fn from(vec: Vec<T>) -> Self {
            Self::from_vec(vec)
        }
    }
    impl<T: Clone> From<&[T]> for List<T> {
        fn from(slice: &[T]) -> Self {
            slice.iter().cloned().collect()
        }
    }
    #[macro_export]
    macro_rules ! list { ( ) => { List :: new ( ) } ; ( $ init : expr ; $ ( $ dim : expr ) ;* ) => { list ! ( $ ( $ dim ) ;* => $ init ) } ; ( $ head : expr ; $ ( $ tail : expr ) ;* => $ init : expr ) => { List :: init ( list ! ( $ ( $ tail ) ;* => $ init ) , $ head ) } ; ( $ head : expr => $ init : expr ) => { List :: init ( $ init , $ head ) } ; }
}
mod prime_number {
    use crate::arraylist::List;
    use crate::counter::Counter;
    pub fn is_prime(n: i64) -> bool {
        for i in (2..).take_while(|i| i * i <= n) {
            if n % i == 0 {
                return false;
            }
        }
        n != 1
    }
    pub fn divisors(n: i64) -> List<i64> {
        let mut ret = List::new();
        for i in (1..).take_while(|i| i * i <= n) {
            if n % i == 0 {
                ret.push(i);
                if i != n / i {
                    ret.push(n / i);
                }
            }
        }
        ret
    }
    pub fn prime_factors(n_: i64) -> Counter<i64> {
        let mut ret = Counter::new();
        let n = std::cell::Cell::new(n_);
        for i in (2..).take_while(|&i| i * i <= n.get()) {
            while n.get() % i == 0 {
                ret[i] += 1;
                n.set(n.get() / i);
            }
        }
        if n.get() != 1 {
            ret[n.get()] = 1;
        }
        ret
    }
    pub fn sieve(n: i32) -> (List<i32>, List<bool>) {
        let mut primes = List::new();
        let mut is_prime = List::init(true, n + 1);
        is_prime[0] = false;
        is_prime[1] = false;
        for i in 2..n + 1 {
            if is_prime[i] {
                primes.push(i);
                for j in (2..).map(|j| j * i).take_while(|&j| j <= n) {
                    is_prime[j] = false;
                }
            }
        }
        (primes, is_prime)
    }
}
mod counter {
    use std::collections::HashMap;
    use std::hash::Hash;
    use std::ops::*;
    #[derive(Clone, Debug)]
    pub struct Counter<K: Eq + Hash> {
        pub cnt: HashMap<K, i64>,
        pub d: i64,
    }
    impl<K: Eq + Hash> Counter<K> {
        pub fn new() -> Counter<K> {
            Counter {
                cnt: HashMap::new(),
                d: 0,
            }
        }
        #[doc = " Remove key when the value <= 0"]
        pub fn dec(&mut self, key: K, delta: i64) {
            if self.by_ref(&key) - delta <= 0 {
                self.remove(&key);
            } else {
                *self.cnt.get_mut(&key).unwrap() -= delta;
            }
        }
        pub fn by_ref(&self, key: &K) -> i64 {
            *self.cnt.get(key).unwrap_or(&self.d)
        }
    }
    impl<K: Eq + Hash> Deref for Counter<K> {
        type Target = HashMap<K, i64>;
        fn deref(&self) -> &Self::Target {
            &self.cnt
        }
    }
    impl<K: Eq + Hash> DerefMut for Counter<K> {
        fn deref_mut(&mut self) -> &mut Self::Target {
            &mut self.cnt
        }
    }
    impl<K: Eq + Hash> Index<K> for Counter<K> {
        type Output = i64;
        fn index(&self, index: K) -> &Self::Output {
            self.cnt.get(&index).unwrap_or(&self.d)
        }
    }
    impl<K: Eq + Hash + Clone> IndexMut<K> for Counter<K> {
        fn index_mut(&mut self, index: K) -> &mut i64 {
            if !self.cnt.contains_key(&index) {
                self.cnt.insert(index.clone(), self.d);
            }
            self.cnt.get_mut(&index).unwrap()
        }
    }
    impl<K: Eq + Hash> std::iter::FromIterator<K> for Counter<K> {
        fn from_iter<T: IntoIterator<Item = K>>(iter: T) -> Self {
            let mut cnt = HashMap::new();
            for i in iter {
                *cnt.entry(i).or_insert(0) += 1;
            }
            Counter { cnt, d: 0 }
        }
    }
    impl<T: Eq + Hash + Clone> Add for Counter<T> {
        type Output = Counter<T>;
        fn add(self, other: Counter<T>) -> Counter<T> {
            let mut ret = Counter::new();
            for (k, v) in self.iter().chain(other.iter()) {
                ret[k.clone()] += *v;
            }
            ret
        }
    }
    impl<T: Eq + Hash + Clone> AddAssign for Counter<T> {
        fn add_assign(&mut self, other: Self) {
            *self = self.clone().add(other);
        }
    }
}
mod range {
    use crate::integer::Int;
    use std::cmp::{max, min};
    use std::ops::{Bound, Range, RangeBounds, RangeInclusive};
    pub trait RangeEx<T> {
        fn width(&self) -> T;
        fn empty(&self) -> bool;
        fn contain_range(&self, inner: &Self) -> bool;
        fn separate_range(&self, other: &Self) -> bool;
        type ReturnRange;
        fn overlap(&self, other: &Self) -> Self::ReturnRange;
    }
    impl<T: Int> RangeEx<T> for Range<T> {
        fn width(&self) -> T {
            if self.empty() {
                T::zero()
            } else {
                self.end - self.start
            }
        }
        fn empty(&self) -> bool {
            !(self.start < self.end)
        }
        fn contain_range(&self, inner: &Self) -> bool {
            self.start <= inner.start && inner.end <= self.end
        }
        fn separate_range(&self, other: &Self) -> bool {
            self.end <= other.start || other.end <= self.start
        }
        type ReturnRange = Range<T>;
        fn overlap(&self, other: &Self) -> Self::ReturnRange {
            let left = max(self.start, other.start);
            let right = min(self.end, other.end);
            left..right
        }
    }
    impl<T: Int> RangeEx<T> for RangeInclusive<T> {
        fn width(&self) -> T {
            if self.empty() {
                T::zero()
            } else {
                *self.end() - *self.start() + T::one()
            }
        }
        fn empty(&self) -> bool {
            !(self.start() <= self.end())
        }
        fn contain_range(&self, inner: &Self) -> bool {
            self.start() <= inner.start() && inner.end() <= self.end()
        }
        fn separate_range(&self, other: &Self) -> bool {
            self.end() <= other.start() || other.end() <= self.start()
        }
        type ReturnRange = RangeInclusive<T>;
        fn overlap(&self, other: &Self) -> Self::ReturnRange {
            let left = *max(self.start(), other.start());
            let right = *min(self.end(), other.end());
            left..=right
        }
    }
    pub trait IntRangeBounds<U: Int>: RangeBounds<U> {
        #[doc = " inclusive"]
        fn lower_bound(&self, lower_bound: U) -> U {
            match self.start_bound() {
                Bound::Included(x) => max(lower_bound, *x),
                Bound::Excluded(x) => max(lower_bound, *x + U::one()),
                Bound::Unbounded => lower_bound,
            }
        }
        #[doc = " exclusive"]
        fn upper_bound(&self, upper_bound: U) -> U {
            match self.end_bound() {
                Bound::Included(x) => min(upper_bound, *x + U::one()),
                Bound::Excluded(x) => min(upper_bound, *x),
                Bound::Unbounded => upper_bound,
            }
        }
        fn to_range(&self, lb: U, ub: U) -> Range<U> {
            self.lower_bound(lb)..self.upper_bound(ub)
        }
    }
    impl<T: ?Sized, U: Int> IntRangeBounds<U> for T where T: RangeBounds<U> {}
}
mod integer {
    pub trait Int:
        std::ops::Add<Output = Self>
        + std::ops::Sub<Output = Self>
        + std::ops::Mul<Output = Self>
        + std::ops::Div<Output = Self>
        + std::ops::Rem<Output = Self>
        + std::hash::Hash
        + PartialEq
        + Eq
        + PartialOrd
        + Ord
        + Copy
    {
        fn to_u8(&self) -> u8;
        fn to_u16(&self) -> u16;
        fn to_u32(&self) -> u32;
        fn to_u64(&self) -> u64;
        fn to_u128(&self) -> u128;
        fn to_i8(&self) -> i8;
        fn to_i16(&self) -> i16;
        fn to_i32(&self) -> i32;
        fn to_i64(&self) -> i64;
        fn to_i128(&self) -> i128;
        fn to_usize(&self) -> usize;
        fn to_isize(&self) -> isize;
        fn zero() -> Self;
        fn one() -> Self;
    }
    macro_rules ! impl_integer_functions { ( $ ( $ name : ident , $ tpe : ident ) ,* ) => { $ ( fn $ name ( & self ) -> $ tpe { * self as $ tpe } ) * } ; }
    macro_rules ! impl_integer { ( $ ( $ tpe : ident ) ,* ) => { $ ( impl Int for $ tpe { impl_integer_functions ! ( to_u8 , u8 , to_u16 , u16 , to_u32 , u32 , to_u64 , u64 , to_u128 , u128 , to_i8 , i8 , to_i16 , i16 , to_i32 , i32 , to_i64 , i64 , to_i128 , i128 , to_usize , usize , to_isize , isize ) ; fn zero ( ) -> Self { 0 } fn one ( ) -> Self { 1 } } ) * } ; }
    impl_integer!(u8, u16, u32, u64, u128, i8, i16, i32, i64, i128, usize, isize);
}