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)

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

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { reader: s.lock() };
    let n: usize = sc.read();
    let s: usize = sc.read();

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

    // x*b+y=n
    // x+y=s
    // x*(b-1)=n-s
    // x<b x<=(b-1)
    // y<b
    if n > s {
        let sq = ((n - s) as f64).sqrt() as usize + 1;
        for x in (1..sq).rev() {
            if (n - s) % x != 0 {
                continue;
            }
            let b = (n - s) / x + 1;
            if n < x * b {
                continue;
            }

            let y = n - x * b;
            if x >= b || y >= b {
                continue;
            }
            assert_eq!(f(b, n), s, "b={} x={} n-s={}", b, x, n - s);
            println!("{}", b);
            return;
        }
    }
    if n == s {
        println!("{}", n + 1);
        return;
    }
    println!("-1");
}

pub struct Scanner<R> {
    reader: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .reader
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}