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 Program
{
    static long n;
    static long s;

    static void Main()
    {
        n = long.Parse(Console.ReadLine());
        s = long.Parse(Console.ReadLine());

        long s_n = (long)Math.Pow(n, 0.5);
        for (long b = 2; b <= s_n; b++)
        {
            long n_ = n;
            long sum = 0;
            while (n_ > 0)
            {
                sum += n_ % b;
                n_ /= b;
            }
            if (s == sum)
            {
                Console.WriteLine(b);
                return;
            }
        }

        long u = (long)((n - s) / (Math.Pow(n, 0.5) - 1));
        for (long k = Math.Min(u, s); k > 0; k--)
        {
            if ((n - s) % k != 0) continue;
            long p = (n + k - s) / k;
            if (k >= p || (s - k) >= p) continue;
            Console.WriteLine(p);
            return;
        }

        if (s == n)
        {
            Console.WriteLine(n + 1);
            return;
        }

        Console.WriteLine(-1);
    }
}