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: Accepted)

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

internal partial class Solver {

    struct Data {
        public int Mod;
        public int LastDigit;
        public int SumDigits;
    }

    public void Run() {
        var k = ni();
        var visited = new bool[k];

        var quFront = new Queue<Data>();
        var quBack = new Queue<Data>();
        quFront.Enqueue(new Data { Mod = 1, SumDigits = 1, LastDigit = 1 });
        while (quFront.Any() || quBack.Any()) {
            var top = quFront.Any() ? quFront.Dequeue() : quBack.Dequeue();
            if (visited[top.Mod]) continue;
            visited[top.Mod] = true;
            if (top.Mod == 0) cout.WriteLine(top.SumDigits);
            if (top.LastDigit < 9)
                quBack.Enqueue(new Data { Mod = (top.Mod + 1) % k, SumDigits = top.SumDigits + 1, LastDigit = top.LastDigit + 1 });
            quFront.Enqueue(new Data { Mod = (top.Mod * 10) % k, SumDigits = top.SumDigits, LastDigit = 0 });
        }
    }

    public class PriorityQueue<T> {
        private T[] array = new T[100];
        private int size = 0;
        private readonly Comparer<T> _comparer;

        public PriorityQueue() {
            _comparer = Comparer<T>.Default;
        }

        public PriorityQueue(Comparison<T> comp) {
            _comparer = Comparer<T>.Create(comp);
        }

        private void Swap(int a, int b) {
            var t = array[a];
            array[a] = array[b];
            array[b] = t;
        }

        private void Expand() {
            var newlist = new T[array.Length * 2];
            Array.Copy(array, newlist, array.Length);
            array = newlist;
        }

        public bool Any() {
            return size > 0;
        }

        public void Enqueue(T newValue) {
            if (size >= array.Length) {
                Expand();
            }
            array[size++] = newValue;
            int pos = size - 1;
            while (pos > 0) {
                int parent = (pos - 1) / 2;
                if (_comparer.Compare(array[parent], array[pos]) > 0) {
                    Swap(parent, pos);
                    pos = parent;
                } else {
                    break;
                }
            }
        }

        public T Peek() {
            return array[0];
        }

        public T Dequeue() {
            var top = array[0];
            array[0] = array[--size];
            int pos = 0;
            while (pos * 2 + 1 < size) {
                int left = pos * 2 + 1;
                int right = left + 1;
                int next = left;
                if (right < size && _comparer.Compare(array[left], array[right]) > 0) {
                    next = right;
                }

                if (_comparer.Compare(array[pos], array[next]) > 0) {
                    Swap(next, pos);
                    pos = next;
                } else {
                    break;
                }
            }

            return top;
        }
    };
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        new Solver(Console.In, Console.Out).Run();
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

    public void RunMulti() {
        while (HasNext()) {
            Run();
        }
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

public static class LinqPadExtension {
    public static T Dump<T>(this T obj) {
#if LOCAL
        return LINQPad.Extensions.Dump(obj);
#else
        return obj;
#endif
    }
}
public class Scanner {
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        var array = new string[size];
        for (var i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        var array = new int[size];
        for (var i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        var array = new long[size];
        for (var i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        var array = new double[size];
        for (var i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (TokenQueue.Count == 0) {
            if (!StockTokens()) {
                throw new InvalidOperationException();
            }
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0) {
            return true;
        }

        return StockTokens();
    }

    private static readonly char[] _separator = new[] { ' ', '\t' };
    private bool StockTokens() {
        while (true) {
            var line = Reader.ReadLine();
            if (line == null) {
                return false;
            }

            var tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) {
                continue;
            }

            foreach (var token in tokens) {
                TokenQueue.Enqueue(token);
            }

            return true;
        }
    }
}