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)

use std::io;
use std::collections::BTreeMap;

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

fn main() {
    let n = read_values::<usize>()[0];
    let s = read_values::<usize>()[0];

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

    for b in 2..400000 {
        if f(b, n) == s {
            println!("{}", b);
            return;
        }
    }

    // x  + y = s
    // bx + y = n
    // => (b-1)x = n - s
    if n <= s {
        println!("-1");
        return;
    }
    let ns = n - s;
    let mut divisors = Vec::new();
    for d in 2..400000 {
        if d * d > ns {
            break;
        }
        if ns % d == 0 {
            divisors.push(d);
            divisors.push(ns / d);
        }
    }
    divisors.push(ns);
    divisors.sort();

    for a in &divisors {
        let b = *a + 1;
        if f(b, n) == s {
            println!("{}", b);
            return;
        }
    }
    println!("-1");
}

fn read_line() -> String {
    let stdin = io::stdin();
    let mut buf = String::new();
    stdin.read_line(&mut buf).unwrap();
    buf
}

fn read_values<T>() -> Vec<T> where T: std::str::FromStr, T::Err: std::fmt::Debug {
    read_line()
        .split(' ')
        .map(|a| a.trim().parse().unwrap())
        .collect()
}