Problem:
Score : 700 points

Problem Statement

Find the smallest possible sum of the digits in the decimal notation of a positive multiple of K.

Constraints

2 \leq K \leq 10^5

K is an integer.

Input

Input is given from Standard Input in the following format:

K

Output

Print the smallest possible sum of the digits in the decimal notation of a positive multiple of K.

Sample Input 1

6

Sample Output 1

3

12=6×2 yields the smallest sum.

Sample Input 2

41

Sample Output 2

5

11111=41×271 yields the smallest sum.

Sample Input 3

79992

Sample Output 3

36

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.IO;
using System.Linq;
using System.Text;
using System.Collections.Generic;
using System.Diagnostics;
using Enu = System.Linq.Enumerable;

public class Program
{
    public void Solve()
    {
        var S = Reader.String();
        int ans = S.Sum(c => c - '0');
        int K = int.Parse(S);
        var dp = Enu.Repeat(int.MaxValue, K + 1).ToArray();
        for (int d = 1; d <= 9; d++) CheckMin(ref dp[d % K], d);

        for (int len = 0; len < 100; len++)
        {
            var nextdp = new int[dp.Length];
            for (int i = 0; i < nextdp.Length; i++) nextdp[i] = int.MaxValue;
            for (int mod = 0, val; mod < K; mod++)
                if ((val = dp[mod]) < int.MaxValue)
                    for (int d = 0; d < 10; d++)
                    {
                        int nextmod = (mod * 10 + d) % K;
                        CheckMin(ref nextdp[nextmod], val + d);
                    }
            dp = nextdp;
            CheckMin(ref ans, dp[0]);
        }

        Console.WriteLine(ans);
    }

    bool CheckMin(ref int a, int b) { if (b < a) { a = b; return true; } return false; }
}



class Entry { static void Main() { new Program().Solve(); } }
class Reader
{
    static TextReader reader = Console.In;
    static readonly char[] separator = { ' ' };
    static readonly StringSplitOptions op = StringSplitOptions.RemoveEmptyEntries;
    static string[] A = new string[0];
    static int i;
    static void Init() { Dispose(); A = new string[0]; }
    public static void Set(TextReader r) { Init(); reader = r; }
    public static void Set(string file) { Init(); reader = new StreamReader(file); }
    public static bool HasNext() { return CheckNext(); }
    public static string String() { return Next(); }
    public static int Int() { return int.Parse(Next()); }
    public static long Long() { return long.Parse(Next()); }
    public static double Double() { return double.Parse(Next()); }
    public static int[] IntLine() { return Array.ConvertAll(Split(Line()), int.Parse); }
    public static int[] IntArray(int N) { return Range(N, Int); }
    public static int[][] IntTable(int H) { return Range(H, IntLine); }
    public static string[] StringArray(int N) { return Range(N, Next); }
    public static string[][] StringTable(int N) { return Range(N, () => Split(Line())); }
    public static string Line() { return reader.ReadLine().Trim(); }
    public static T[] Range<T>(int N, Func<T> f) { var r = new T[N]; for (int i = 0; i < N; r[i++] = f()) ; return r; }
    public static void Dispose() { reader.Dispose(); }
    static string[] Split(string s) { return s.Split(separator, op); }
    static string Next() { CheckNext(); return A[i++]; }
    static bool CheckNext()
    {
        if (i < A.Length) return true;
        string line = reader.ReadLine();
        if (line == null) return false;
        if (line == "") return CheckNext();
        A = Split(line);
        i = 0;
        return true;
    }
}