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_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;

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()
}

fn func(b: u64, n: u64) -> u64 {
    if n < b {
        return n;
    }
    func(b, n / b) + n % b
}

fn solve() {
    let n: u64 = read();
    let s: u64 = read();
    if s > n {
        println!("-1");
        return;
    }
    if s == n {
        println!("{}", n + 1);
        return;
    }
    for b in 2..((n as f64).sqrt() as u64 + 1) {
        if func(b, n) == s {
            println!("{}", b);
            return;
        }
    }
    for p in (1..((n as f64).sqrt() as u64 + 1)).rev() {
        if (n - s) % p != 0 {
            continue;
        }
        let b = (n - s) / p + 1;
        if func(b, n) == s {
            println!("{}", b);
            return;
        }
    }
    println!("-1");
}

fn main() {
    std::thread::Builder::new()
        .name("big stack size".into())
        .stack_size(32 * 1024 * 1024) // 32 MBのスタックサイズ
        .spawn(|| {solve()})
        .unwrap()
        .join()
        .unwrap();
}