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.Collections.Generic;
using System.Linq;
using System.IO;
using System.Runtime.CompilerServices;
using System.Text;
using static Template;
using static System.Console;
using static System.Convert;
using static System.Math;
using Pi = Pair<int, int>;
using Graph;
//using DataStructure;

class Solver
{
    public void Solve()
    {
        var k = Input.num;
        var g = new BreadthFirstSearch_01(k);
        for (var i = 0; i < k; i++)
        {
            if (i % 10 != 9)
                g.AddEdge(i, (i + 1) % k, 1);
            g.AddEdge(i, 10 * i % k, 0);
        }
        WriteLine(g.Execute(1)[0] + 1);
    }
}

namespace Graph
{
    #region テンプレート      
    public class Graph<E> where E : Edge
    {
        public List<E>[] Edges { get; set; }
        public int Count { get; }
        public Graph(int count)
        {
            Count = count;
            Edges = Create(count, () => new List<E>());
        }
    }
    public class Directed<E> : Graph<E> where E : Edge
    { public Directed(int count) : base(count) { } }
    public class Undirected<E> : Graph<E> where E : Edge
    { public Undirected(int count) : base(count) { } }
    public interface IWeight<T> where T : IComparable<T> { T Weight { get; set; } }
    public interface INNegWeight<T> : IWeight<T> where T : IComparable<T> { }
    public interface IVertex
    {
        int Id { get; }
    }
    public class Edge
    {
        public int From { get; set; }
        public int To { get; set; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public Edge(int from, int to)
        { From = from; To = to; }
    }
    #endregion
}
namespace Graph
{
    #region 01BFS
    /// <summary>
    /// 依存:Deque
    /// </summary>
    public class BreadthFirstSearch_01
    {
        private Graph<BfsEdge> g;
        public BreadthFirstSearch_01(int count)
        { g = new Graph<BfsEdge>(count); }
        public void AddEdge(int from, int to, int weight)
            => g.Edges[from].Add(new BfsEdge(from, to, weight));
        /// <summary>
        /// O(V+E)
        /// </summary>
        /// <param name="st"></param>
        /// <returns></returns>
        public int[] Execute(int st = 0)
        {
            var deq = new Deque<int>();
            var dist = Create(g.Count, () => -1);
            dist[st] = 0;
            deq.EnqueueHead(st);
            while(deq.Count!=0)
            {
                var p = deq.DequeueHead();
                foreach (var e in g.Edges[p])
                    if (dist[e.To] == -1)
                    {
                        dist[e.To] = dist[p] + e.Weight;
                        if (e.Weight == 1)
                            deq.EnqueueTail(e.To);
                        else deq.EnqueueHead(e.To);
                    }
            }
            return dist;
        }
        
        public class BfsEdge:Edge,IWeight<int>
        {
            public int Weight { get; set; }
            public BfsEdge(int from,int to,int weight):base(from,to)
            { Weight = weight; }
        }
    }
    public class BreadthFirstSearch_01<V> where V : IVertex
    {
        private BreadthFirstSearch_01 g;
        public BreadthFirstSearch_01(int count)
        { g = new BreadthFirstSearch_01(count); }
        public void AddEdge(V from, V to, int weight)
            => g.AddEdge(from.Id, to.Id, weight);
        /// <summary>
        /// O(V+E)
        /// </summary>
        /// <param name="st"></param>
        /// <returns></returns>
        public int[] Execute(V st)
            => g.Execute(st.Id);
    }
    #endregion
}

public class Deque<T>
{
    private RingBuffer<T> _buf;
    int _offset = 0, _size;
    public int Count { get; private set; }
    public T PeekHead { get { return _buf[_offset]; } }
    public T PeekTail { get { return _buf[Count + _offset - 1]; } }
    public Deque() { _buf = new RingBuffer<T>(_size = 16); }
    public Deque(int size) { _buf = new RingBuffer<T>(_size = size); }
    public T this[int index]
    {
        get { return _buf[index + _offset]; }
        set { _buf[index + _offset] = value; }
    }
    private void Extend()
    {
        var t = new RingBuffer<T>(_size << 1);
        for (var i = 0; i < _size; i++)
            t[i] = _buf[_offset + i];
        _offset = 0;
        _buf = t;
        _size <<= 1;
    }
    public void EnqueueHead(T item)
    {
        if (Count == _size) Extend();
        _buf[--_offset] = item;
        Count++;
    }
    public T DequeueHead()
    {
        if (Count == 0) return default(T);
        Count--;
        return _buf[_offset++];
    }
    public void EnqueueTail(T item)
    {
        if (Count == _size) Extend();
        _buf[Count++ + _offset] = item;
    }
    public T DequeueTail()
    {
        if (Count == 0) return default(T);
        return _buf[--Count + _offset];
    }
}

public class RingBuffer<T>
{
    private T[] _item;
    public int Count { get; private set; }
    public RingBuffer(int size)
    {
        _item = new T[Count = size];
    }
    public T this[int index]
    {
        get { index %= Count; if (index < 0) index += Count; return _item[index]; }
        set { index %= Count; if (index < 0) index += Count; _item[index] = value; }
    }
}

public class Template
{
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == 1) { num = val; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == -1) { num = val; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T v1, ref T v2)
    { var t = v2; v2 = v1; v1 = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<T> f)
        => Enumerable.Repeat(0, n).Select(_ => f()).ToArray();
    public static void Fail() => Fail("No");
    public static void Fail<T>(T s) { WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        var p = new Solver();
        for (var i = 1; i > 0; --i)
            p.Solve();
        Console.Out.Flush();
    }
}

public class Input
{
    public static string read => ReadLine().Trim();
    public static int[] ar => read.Split(' ').Select(int.Parse).ToArray();
    public static int num => ToInt32(read);
    public static long[] arL => read.Split(' ').Select(long.Parse).ToArray();
    public static long numL => ToInt64(read);
    public static char[][] grid(int h)
        => Create(h, () => read.ToCharArray());
    public static int[] ar1D(int n)
        => Create(n, () => num);
    public static long[] arL1D(int n)
        => Create(n, () => numL);
    public static string[] strs(int n)
        => Create(n, () => read);
    public static int[][] ar2D(int n)
        => Create(n, () => ar);
    public static long[][] arL2D(int n)
        => Create(n, () => arL);
    public static List<T>[] edge<T>(int n)
        => Create(n, () => new List<T>());
    public static void Make<T1, T2>(out T1 v1, out T2 v2)
    {
        v1 = Next<T1>();
        v2 = Next<T2>();
    }
    public static void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3)
    {
        Make(out v1, out v2);
        v3 = Next<T3>();
    }
    public static void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4)
    {
        Make(out v1, out v2, out v3);
        v4 = Next<T4>();
    }
    public static void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5)
    {
        Make(out v1, out v2, out v3, out v4);
        v5 = Next<T5>();
    }
    public static void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6)
    {
        Make(out v1, out v2, out v3, out v4, out v5);
        v6 = Next<T6>();
    }
    static Input()
    {
        sc = new Queue<string>();
        dic = new Dictionary<Type, Func<string, object>>();
        dic[typeof(int)] = s => int.Parse(s);
        dic[typeof(long)] = s => long.Parse(s);
        dic[typeof(char)] = s => char.Parse(s);
        dic[typeof(double)] = s => double.Parse(s);
        dic[typeof(uint)] = s => uint.Parse(s);
        dic[typeof(ulong)] = s => ulong.Parse(s);
        dic[typeof(string)] = s => s;
    }
    private static Dictionary<Type, Func<string, object>> dic;
    private static Queue<string> sc;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T Next<T>() { if (sc.Count == 0) foreach (var item in read.Split(' ')) sc.Enqueue(item); return (T)dic[typeof(T)](sc.Dequeue()); }
    public const int MOD = 1000000007;
}

public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1 { get; set; }
    public T2 v2 { get; set; }
    public Pair() { v1 = Input.Next<T1>(); v2 = Input.Next<T2>(); }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString()
        => $"{v1.ToString()} {v2.ToString()}";
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3 { get; set; }
    public Pair() : base() { v3 = Input.Next<T3>(); }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString()
        => $"{base.ToString()} {v3.ToString()}";
}