Problem:
Score : 500 points

Problem Statement

For integers b (b \geq 2) and n (n \geq 1), let the function f(b,n) be defined as follows:

f(b,n) = n, when n < b

f(b,n) = f(b,\,{\rm floor}(n / b)) + (n \ {\rm mod} \ b), when n \geq b

Here, {\rm floor}(n / b) denotes the largest integer not exceeding n / b,
and n \ {\rm mod} \ b denotes the remainder of n divided by b.

Less formally, f(b,n) is equal to the sum of the digits of n written in base b.
For example, the following hold:

f(10,\,87654)=8+7+6+5+4=30

f(100,\,87654)=8+76+54=138

You are given integers n and s.
Determine if there exists an integer b (b \geq 2) such that f(b,n)=s.
If the answer is positive, also find the smallest such b.

Constraints

1 \leq n \leq 10^{11}

1 \leq s \leq 10^{11}

n,\,s are integers.

Input

The input is given from Standard Input in the following format:

n
s

Output

If there exists an integer b (b \geq 2) such that f(b,n)=s, print the smallest such b.
If such b does not exist, print -1 instead.

Sample Input 1

87654
30

Sample Output 1

10

Sample Input 2

87654
138

Sample Output 2

100

Sample Input 3

87654
45678

Sample Output 3

-1

Sample Input 4

31415926535
1

Sample Output 4

31415926535

Sample Input 5

1
31415926535

Sample Output 5

-1

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_macros)]
macro_rules! debug {
    ($($arg:tt)*) => (if cfg!(debug_assertions) {print!("[DEBUG] "); println!($($arg)*)});
}

fn f(b: i64, n: i64) -> i64 {
    if n < b {
        n
    } else {
        f(b, n / b) + n % b
    }
}

const N: i64 = 1000000;

fn main() {
    let mut rd = reader::Reader::new();

    let n: i64 = rd.next();
    let s: i64 = rd.next();

    if n == s {
        println!("{}", n + 1);
        return;
    }

    for i in 2..N {
        if f(i, n) == s {
            println!("{}", i);
            return;
        }
    }

    let mut ans: i64 = -1;
    for i in 1..N {
        let b = n / i;
        if b < 2 || i * i > n {
            break;
        }
        let r = n % b;
        let nb = n / (i + 1) + 1;

        debug!("i = {}, b = {}, r = {}, nb = {}", i, b, r, nb);
        let lb = f(b, n);
        let ub = f(nb, n);

        debug!("lb = {}, ub = {}", lb, ub);
        if lb <= s && s <= ub && (s - n % b) % i == 0 {
            ans = b - (s - i) / i;
            debug!("cand = {}, f(n, b) = {}", ans, f(ans, n));
        }
    }
    println!("{}", ans);
}

mod reader {
    use std::fmt::Debug;
    use std::str::FromStr;
    use std::collections::VecDeque;
    use std::io::{self, Read};

    fn _read_line() -> String {
        let mut buf = String::new();
        let _ = io::stdin().read_to_string(&mut buf);
        buf
    }

    fn _read_vec_deque_string() -> VecDeque<String> {
        _read_line()
            .split_whitespace()
            .map(|s| s.to_string())
            .collect()
    }

    #[allow(dead_code)]
    pub struct Reader {
        buf: VecDeque<String>,
    }

    #[allow(dead_code)]
    impl Reader {
        pub fn new() -> Reader {
            Reader { buf: VecDeque::new() }
        }

        fn update_buf(&mut self) {
            if self.buf.is_empty() {
                self.buf = _read_vec_deque_string();
            }
        }

        pub fn next<T>(&mut self) -> T
            where
                T: FromStr,
                T::Err: Debug,
        {
            self.update_buf();
            self.buf.pop_front().unwrap().parse::<T>().unwrap()
        }

        pub fn next_n<T>(&mut self, n: usize) -> Vec<T>
            where
                T: FromStr,
                T::Err: Debug,
        {
            let mut vec: Vec<T> = Vec::with_capacity(n);

            for _ in 0..n {
                vec.push(self.next());
            }
            vec
        }
    }
}