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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;

class D {
    static long F(long b, long n) {
        var answer = 0L;
        while (n != 0L) {
            answer += n % b;
            n /= b;
        }
        return answer;
    }

    static long Solve(long n, long s) {
        if (n == s) return n + 1L;
        var sqrtN = (long)Math.Round(Math.Sqrt(n));
        var sqrtNLower = sqrtN;
        if (n < sqrtNLower * sqrtNLower) --sqrtNLower;
        var sqrtNUpper = sqrtN;
        if (sqrtNUpper * sqrtNUpper < n) ++sqrtNUpper;
        for (long b = 2L; b < sqrtNUpper; ++b)
            if (F(b, n) == s) return b;
        for (long s1 = sqrtNLower; s1 != 0L; --s1) {
            var s2 = s - s1;
            var b = (n - s2) / s1;
            if (b >= 2L && F(b, n) == s) return b;
        }
        return -1L;
    }

    public static void Main() {
        var n = long.Parse(Console.ReadLine());
        var s = long.Parse(Console.ReadLine());
        Console.WriteLine(Solve(n, s));
    }
}