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;
using System.Collections.Generic;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading;

// (づ°ω°)づﾐe★゜・。。・゜゜・。。・゜☆゜・。。・゜゜・。。・゜
public class Solver
{
    const int LIMIT = 1000000;

    public void Solve()
    {
        long n = ReadLong();
        long m = ReadLong();

        if (n < m)
        {
            Write(-1);
            return;
        }
        else if (n == m)
        {
            Write(n + 1);
            return;
        }

        for (int i = 2; i <= LIMIT; i++)
        {
            long s = 0;
            long x = n;
            while (x > 0)
            {
                s += x % i;
                x /= i;
            }
            if (s == m)
            {
                Write(i);
                return;
            }
        }

        long ans = long.MaxValue;
        for (int i = 1; i <= m && 1L * i * LIMIT <= n; i++)
        {
            if ((n + i - m) % i == 0)
            {
                long v = (n + i - m) / i;
                if (i < v && m - i < v)
                    ans = Math.Min(ans, v);
            }
        }
        Write(ans < long.MaxValue ? ans : -1);
    }

    #region Main

    protected static TextReader reader;
    protected static TextWriter writer;
    static void Main()
    {
#if DEBUG
        reader = new StreamReader("..\\..\\input.txt");
        //reader = new StreamReader(Console.OpenStandardInput());
        writer = Console.Out;
        //writer = new StreamWriter("..\\..\\output.txt");
#else
        reader = new StreamReader(Console.OpenStandardInput());
        writer = new StreamWriter(Console.OpenStandardOutput());
        //reader = new StreamReader("input.txt");
        //writer = new StreamWriter("output.txt");
#endif
        try
        {
            new Solver().Solve();
            //var thread = new Thread(new Solver().Solve, 1024 * 1024 * 128);
            //thread.Start();
            //thread.Join();
        }
        catch (Exception ex)
        {
#if DEBUG
            Console.WriteLine(ex);
#else
            throw;
#endif
        }
        reader.Close();
        writer.Close();
    }

    #endregion

    #region Read / Write
    private static Queue<string> currentLineTokens = new Queue<string>();
    private static string[] ReadAndSplitLine() { return reader.ReadLine().Split(new[] { ' ', '\t', }, StringSplitOptions.RemoveEmptyEntries); }
    public static string ReadToken() { while (currentLineTokens.Count == 0)currentLineTokens = new Queue<string>(ReadAndSplitLine()); return currentLineTokens.Dequeue(); }
    public static int ReadInt() { return int.Parse(ReadToken()); }
    public static long ReadLong() { return long.Parse(ReadToken()); }
    public static double ReadDouble() { return double.Parse(ReadToken(), CultureInfo.InvariantCulture); }
    public static int[] ReadIntArray() { return ReadAndSplitLine().Select(int.Parse).ToArray(); }
    public static long[] ReadLongArray() { return ReadAndSplitLine().Select(long.Parse).ToArray(); }
    public static double[] ReadDoubleArray() { return ReadAndSplitLine().Select(s => double.Parse(s, CultureInfo.InvariantCulture)).ToArray(); }
    public static int[][] ReadIntMatrix(int numberOfRows) { int[][] matrix = new int[numberOfRows][]; for (int i = 0; i < numberOfRows; i++)matrix[i] = ReadIntArray(); return matrix; }
    public static int[][] ReadAndTransposeIntMatrix(int numberOfRows)
    {
        int[][] matrix = ReadIntMatrix(numberOfRows); int[][] ret = new int[matrix[0].Length][];
        for (int i = 0; i < ret.Length; i++) { ret[i] = new int[numberOfRows]; for (int j = 0; j < numberOfRows; j++)ret[i][j] = matrix[j][i]; } return ret;
    }
    public static string[] ReadLines(int quantity) { string[] lines = new string[quantity]; for (int i = 0; i < quantity; i++)lines[i] = reader.ReadLine().Trim(); return lines; }
    public static void WriteArray<T>(IEnumerable<T> array) { writer.WriteLine(string.Join(" ", array)); }
    public static void Write(params object[] array) { WriteArray(array); }
    public static void WriteLines<T>(IEnumerable<T> array) { foreach (var a in array)writer.WriteLine(a); }
    private class SDictionary<TKey, TValue> : Dictionary<TKey, TValue>
    {
        public new TValue this[TKey key]
        {
            get { return ContainsKey(key) ? base[key] : default(TValue); }
            set { base[key] = value; }
        }
    }
    private static T[] Init<T>(int size) where T : new() { var ret = new T[size]; for (int i = 0; i < size; i++)ret[i] = new T(); return ret; }
    #endregion
}