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;

public class Input
{
    public long n;

    public long s;

    public Input()
    {
        string[] input = null;

        // スペース区切りの整数の入力(横、個別)
        input = Console.ReadLine().Split(' ');

        n = long.Parse(input[0]);

        // スペース区切りの整数の入力(横、個別)
        input = Console.ReadLine().Split(' ');

        s = long.Parse(input[0]);
    }
}

public class Program
{
    public static void Main(string[] args)
    {
        string output = Proc();

        // 出力
        Console.WriteLine(output);
    }

    static string Proc()
    {
        Input input = new Input();
        string output = string.Empty;

        long result = -1;

        // n = s
        if (input.n == input.s)
        {
            result = input.n + 1;
            output = result.ToString();
            return output;
        }

        // n < s
        if (input.n < input.s)
        {
            result = -1;
            output = result.ToString();
            return output;
        }

        // rootNまで
        long rootN = (long)Math.Sqrt(input.n);

        for (long b = 2; b <= rootN; b++)
        {
            long s = DigitSum(b, input.n);

            if (s == input.s)
            {
                result = b;
                output = result.ToString();
                return output;
            }
        }

        // rootN以降(2桁以下)

        // n = pb + q (1)
        // p + q = s  (2)

        // n = pb + (s - p)
        // pb = n - s + p
        // b = (n - s + p) / p
        // b = (n - s) / p + 1

        for (long p = rootN; p >= 1; p--)
        {
            long b = (input.n - input.s) / p + 1;
            if (b <= 1)
            {
                continue;
            }

            long s = DigitSum(b, input.n);

            if (s == input.s)
            {
                result = b;
                output = result.ToString();
                return output;
            }
        }

        output = result.ToString();
        return output;
    }

    static long DigitSum(long b, long n)
    {
        if (b > n)
        {
            return n;
        }

        long result = DigitSum(b, n / b);
        result += n % b;

        return result;
    }
}