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: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.IO;

class Scanner
{
    private string[] Line;
    private int Index;

    public Scanner()
    {
        Line = new string[0];
        Index = 0;
    }

    public string Next()
    {
        if (Index >= Line.Length)
        {
            Line = Console.ReadLine().Split(' ');
            Index = 0;
        }
        var ret = Line[Index];
        Index++;
        return ret;
    }

    public int NextInt()
    {
        return int.Parse(Next());
    }
}

struct S
{
    public int Num, Sum;

    public S(int num, int sum)
    {
        this.Num = num;
        this.Sum = sum;
    }
}

class Program
{
    private int K;

    private void Scan()
    {
        var sc = new Scanner();
        K = sc.NextInt();
    }

    public void Solve()
    {
        Scan();
        var ans = new int[K];
        var q = new LinkedList<S>();
        q.AddLast(new S(1, 1));
        while (q.Count > 0)
        {
            var t = q.Last();
            q.RemoveLast();
            if (ans[t.Num] != 0)
            {
                continue;
            }
            ans[t.Num] = t.Sum;
            q.AddLast(new S((t.Num * 10) % K, t.Sum));
            q.AddFirst(new S((t.Num + 1) % K, t.Sum + 1));
        }
        Console.WriteLine(ans[0]);
    }

    static public void Main()
    {
        new Program().Solve();
    }
}