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;
using System.Collections.Generic;
using System.Collections.Specialized;
using System.Text;
using System.Text.RegularExpressions;
using System.Linq;
using System.Numerics;
using System.IO;
public class CodeForces
{
#if TEST
// To set this go to Project -> Properties -> Build -> General -> Conditional compilation symbols: -> enter 'TEST' into text box.
const bool testing = true;
#else
    const bool testing = false;
#endif

    public abstract class Heap<T> : IEnumerable<T>
    {
        private const int InitialCapacity = 0;
        private const int GrowFactor = 2;
        private const int MinGrow = 1;

        private int _capacity = InitialCapacity;
        private T[] _heap = new T[InitialCapacity];
        private int _tail = 0;

        public int Count { get { return _tail; } }
        public int Capacity { get { return _capacity; } }

        protected Comparer<T> Comparer { get; private set; }
        protected abstract bool Dominates(T x, T y);

        protected Heap() : this(Comparer<T>.Default)
        {
        }

        protected Heap(Comparer<T> comparer) : this(Enumerable.Empty<T>(), comparer)
        {
        }

        protected Heap(IEnumerable<T> collection)
            : this(collection, Comparer<T>.Default)
        {
        }

        protected Heap(IEnumerable<T> collection, Comparer<T> comparer)
        {
            if (collection == null) throw new ArgumentNullException("collection");
            if (comparer == null) throw new ArgumentNullException("comparer");

            Comparer = comparer;

            foreach (var item in collection)
            {
                if (Count == Capacity)
                    Grow();

                _heap[_tail++] = item;
            }

            for (int i = Parent(_tail - 1); i >= 0; i--)
                BubbleDown(i);
        }

        public void Add(T item)
        {
            if (Count == Capacity)
                Grow();

            _heap[_tail++] = item;
            BubbleUp(_tail - 1);
        }

        private void BubbleUp(int i)
        {
            if (i == 0 || Dominates(_heap[Parent(i)], _heap[i]))
                return; //correct domination (or root)

            Swap(i, Parent(i));
            BubbleUp(Parent(i));
        }

        public T GetMin()
        {
            if (Count == 0) throw new InvalidOperationException("Heap is empty");
            return _heap[0];
        }

        public T ExtractDominating()
        {
            if (Count == 0) throw new InvalidOperationException("Heap is empty");
            T ret = _heap[0];
            _tail--;
            Swap(_tail, 0);
            BubbleDown(0);
            return ret;
        }

        private void BubbleDown(int i)
        {
            int dominatingNode = Dominating(i);
            if (dominatingNode == i) return;
            Swap(i, dominatingNode);
            BubbleDown(dominatingNode);
        }

        private int Dominating(int i)
        {
            int dominatingNode = i;
            dominatingNode = GetDominating(YoungChild(i), dominatingNode);
            dominatingNode = GetDominating(OldChild(i), dominatingNode);

            return dominatingNode;
        }

        private int GetDominating(int newNode, int dominatingNode)
        {
            if (newNode < _tail && !Dominates(_heap[dominatingNode], _heap[newNode]))
                return newNode;
            else
                return dominatingNode;
        }

        private void Swap(int i, int j)
        {
            T tmp = _heap[i];
            _heap[i] = _heap[j];
            _heap[j] = tmp;
        }

        private static int Parent(int i)
        {
            return (i + 1) / 2 - 1;
        }

        private static int YoungChild(int i)
        {
            return (i + 1) * 2 - 1;
        }

        private static int OldChild(int i)
        {
            return YoungChild(i) + 1;
        }

        private void Grow()
        {
            int newCapacity = _capacity * GrowFactor + MinGrow;
            var newHeap = new T[newCapacity];
            Array.Copy(_heap, newHeap, _capacity);
            _heap = newHeap;
            _capacity = newCapacity;
        }

        public IEnumerator<T> GetEnumerator()
        {
            return _heap.Take(Count).GetEnumerator();
        }

        IEnumerator IEnumerable.GetEnumerator()
        {
            return GetEnumerator();
        }
    }

    public class MaxHeap<T> : Heap<T>
    {
        public MaxHeap()
            : this(Comparer<T>.Default)
        {
        }

        public MaxHeap(Comparer<T> comparer)
            : base(comparer)
        {
        }

        public MaxHeap(IEnumerable<T> collection, Comparer<T> comparer)
            : base(collection, comparer)
        {
        }

        public MaxHeap(IEnumerable<T> collection) : base(collection)
        {
        }

        protected override bool Dominates(T x, T y)
        {
            return Comparer.Compare(x, y) >= 0;
        }
    }

    public class MinHeap<T> : Heap<T>
    {
        public MinHeap()
            : this(Comparer<T>.Default)
        {
        }

        public MinHeap(Comparer<T> comparer)
            : base(comparer)
        {
        }

        public MinHeap(IEnumerable<T> collection) : base(collection)
        {
        }

        public MinHeap(IEnumerable<T> collection, Comparer<T> comparer)
            : base(collection, comparer)
        {
        }

        protected override bool Dominates(T x, T y)
        {
            return Comparer.Compare(x, y) <= 0;
        }
    }
    class Node : IComparable
    {
        public int Index { get; set; }
        public int Value { get; set; }
        public int CompareTo(object obj)
        {
            return this.Value.CompareTo(((Node)obj).Value);
        }
    }

    public static void Main(string[] args)
    {
        var k = int.Parse(Console.ReadLine().TrimEnd());
        var heap = new MinHeap<Node>();
        heap.Add(new Node()
        {
            Index = 1,
            Value = 1
        });
        var visited = new bool[k];
        visited[1] = true;
        
        while(heap.Count > 0)
        {
            var cur = heap.ExtractDominating();
            if(cur.Index == 0)
            {
                Console.WriteLine(cur.Value);
                return;
            }
            if (!visited[(cur.Index * 10)%k])
            {
                visited[(cur.Index * 10) % k] = true;
                heap.Add(new Node()
                {
                    Index = (cur.Index * 10) % k,
                    Value = cur.Value
                });
            }
            if (!visited[(cur.Index + 1) % k])
            {
                visited[(cur.Index + 1) % k] = true;
                heap.Add(new Node()
                {
                    Index = (cur.Index + 1) % k,
                    Value = cur.Value + 1
                });
            }
        }

    }
}