Problem:
Score : 300 points

Problem Statement

You are given an integer N.

For two positive integers A and B, we will define F(A,B) as the larger of the following: the number of digits in the decimal notation of A, and the number of digits in the decimal notation of B.

For example, F(3,11) = 2 since 3 has one digit and 11 has two digits.

Find the minimum value of F(A,B) as (A,B) ranges over all pairs of positive integers such that N = A \times B.

Constraints

1 \leq N \leq 10^{10}

N is an integer.

Input

The input is given from Standard Input in the following format:

N

Output

Print the minimum value of F(A,B) as (A,B) ranges over all pairs of positive integers such that N = A \times B.

Sample Input 1

10000

Sample Output 1

3

F(A,B) has a minimum value of 3 at (A,B)=(100,100).

Sample Input 2

1000003

Sample Output 2

7

There are two pairs (A,B) that satisfy the condition: (1,1000003) and (1000003,1). For these pairs, F(1,1000003)=F(1000003,1)=7.

Sample Input 3

9876543210

Sample Output 3

6

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

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

namespace ABC057C
{
    class Program
    {
        static void Solve(Input input)
        {
            var n = input.NextLong();

            // nを因数分解した結果をいい感じにレイアウトする
            var list = Prime.PrimeFactors(n).ToArray();
            long min = int.MaxValue;
            for (long i = 0; i < 1L << list.Length; i++)
            {
                long a = 1;
                long b = 1;

                for (int j = 0; j < list.Length; j++)
                {
                    if ((i & (1L << j)) == 0)
                    {
                        a *= list[j];
                    }
                    else
                    {
                        b *= list[j];
                    }
                }

                min = Math.Min(min, Math.Max(a, b));
            }

            Console.WriteLine(min.ToString().Length);

        }

        /// <summary>
        /// 素数判定
        /// 素因数分解
        /// 素数列挙（エラトステネスの篩）
        /// 約数列挙
        /// </summary>
        static class Prime
        {
            /// <summary>
            /// 素数か
            /// </summary>
            /// <param name="x"></param>
            /// <returns></returns>
            public static bool IsPrime(long x)
            {
                if (x < 2)
                {
                    return false;
                }
                if (x == 2)
                {
                    return true;
                }
                if ((x % 2) == 0)
                {
                    return false;
                }

                long rx = (long)(Math.Sqrt(x) + 1);
                for (long n = 3; n < rx; n += 2)
                {
                    if ((x % n) == 0)
                    {
                        return false;
                    }
                }

                return true;
            }

            /// <summary>
            /// 素数判定リスト
            /// </summary>
            /// <param name="n"></param>
            /// <returns></returns>
            public static bool[] PrimeSieve(int n)
            {
                // Sieve of Eratosthenes
                var sieve = Enumerable.Repeat<bool>(true, n + 1).ToArray();

                if (n > 2)
                {
                    sieve[0] = sieve[1] = false;
                }

                var rn = (int)(Math.Sqrt(n) + 1);
                foreach (var i in Enumerable.Range(2, rn - 2))
                {
                    if (sieve[i])
                    {
                        for (int j = 2 * i; j <= n; j += i)
                        {
                            sieve[j] = false;
                        }
                    }
                }

                return sieve;
            }

            /// <summary>
            /// 素数リスト
            /// </summary>
            /// <param name="n"></param>
            /// <returns></returns>
            public static IEnumerable<int> PrimeList(int n)
            {
                var sieve = PrimeSieve(n);

                return sieve.Select((x, i) => new { X = x, I = i })
                    .Where(x => x.X)
                    .Select(x => x.I);
            }

            /// <summary>
            /// 素因数分解
            /// </summary>
            /// <param name="n"></param>
            /// <returns></returns>
            public static IEnumerable<long> PrimeFactors(long n)
            {
                var m = n;
                var m2 = (long)Math.Sqrt(m) + 1;
                var p = 2;

                while (p < m2)
                {
                    if ((m % p) == 0)
                    {
                        m = m / p;
                        m2 = (long)Math.Sqrt(m) + 1;
                        yield return p;
                    }
                    else
                    {
                        p++;
                    }
                }
                if (m > 1)
                {
                    yield return m;
                }
            }

            /// <summary>
            /// n以下でnと互いに素な数の個数
            /// オイラーのファイ関数
            /// </summary>
            /// <param name="n"></param>
            /// <returns></returns>
            public static long PrimeCount(long n)
            {
                var primes = PrimeFactors(n).Distinct();

                var dsum = new decimal(1);
                foreach (var p in primes)
                {
                    var pp = new decimal(p);

                    dsum *= 1 - (1 / pp);
                }

                return (long)Math.Round(dsum * n);
            }

            /// <summary>
            /// 約数列挙 順番はバラバラなので注意！
            /// </summary>
            /// <param name="n"></param>
            /// <returns></returns>
            static IEnumerable<long> Divisors(long n)
            {
                if (n < 1)
                {
                    yield break;
                }

                var rn = (int)(Math.Sqrt(n) + 1);
                for (long i = 1; i < rn; i++)
                {
                    if ((n % i) == 0)
                    {
                        yield return i;

                        if (i != (n / i))
                        {
                            yield return n / i;
                        }
                    }
                }
            }

            /// <summary>
            /// 約数列挙
            /// </summary>
            /// <param name="n"></param>
            /// <param name="needsSort">昇順にソートするか</param>
            /// <returns></returns>
            public static IEnumerable<long> Divisors(long n, bool needsSort)
            {
                var divs = Divisors(n);

                if (needsSort)
                {
                    return divs.OrderBy(x => x);
                }
                else
                {
                    return divs;
                }
            }
        }

        #region Competitive Template
        static readonly int MOD = (int)1e9 + 7;

        public static void Main(string[] args)
        {
            // 出力が少ないときはこれをセットする方が時間かかるけれど
            // その場合は出力がボトルネックになっていないので、まあ良しとする
            var needsFlushOutput = true;
            if (needsFlushOutput)
            {
                // 細かく出力しないようにする
                var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
                Console.SetOut(sw);
            }

            // 通常は引数無しで、ファイルから読み出したい場合はpath指定する
            var input = new Input();
            // 仮想的に標準入力をセットする
            //input.SetText("");

            Solve(input);

            Console.Out.Flush();
        }

        class Input
        {
            // 変な改行コードがたまに混じっているので、一応セパレート指定する
            // スペース単独指定の方がもちろん早い
            static readonly char[] separator = { ' ', '\r', '\n' };
            Queue<string> q { get; set; }
            StreamReader sr;

            /// <summary>
            /// 特定のファイルから読み出したい場合はpath指定する
            /// </summary>
            /// <param name="path"></param>
            public Input(string path = "")
            {
                q = new Queue<string>();

                if (string.IsNullOrEmpty(path))
                {
                    sr = new StreamReader(Console.OpenStandardInput());
                }
                else
                {
                    sr = new StreamReader(path);
                }
            }

            /// <summary>
            /// 入力予約
            /// </summary>
            /// <param name="items"></param>
            public void SetText(IEnumerable<string> items)
            {
                foreach (var item in items)
                {
                    SetText(item);
                }
            }

            /// <summary>
            /// 入力予約
            /// </summary>
            /// <param name="s"></param>
            /// <returns></returns>
            public bool SetText(string s)
            {
                if (s == null) return false;

                foreach (var elem in s.Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries))
                {
                    q.Enqueue(elem);
                }

                return true;
            }

            /// <summary>
            /// 内部queueに入力からの値をsplitして格納する
            /// </summary>
            /// <returns></returns>
            bool read()
            {
                var s = sr.ReadLine();
                if (s == null) return false;

                foreach (var elem in s.Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries))
                {
                    q.Enqueue(elem);
                }

                if (!q.Any()) return read();

                return true;
            }

            /// <summary>
            /// 次のstringを一つ読み込む
            /// </summary>
            /// <returns></returns>
            public string Next()
            {
                if (!q.Any())
                {
                    if (!read())
                    {
                        return "";
                    }
                }

                return q.Dequeue();
            }

            /// <summary>
            /// 指定個数queueにたまるまでenqueueし続ける
            /// </summary>
            /// <param name="n"></param>
            /// <returns></returns>
            bool accumulate(int n)
            {
                while (q.Count() < n)
                {
                    if (!read()) return false;
                }

                return true;
            }

            public int NextInt() => int.Parse(Next());
            public long NextLong() => long.Parse(Next());
            public double NextDouble() => double.Parse(Next());

            /// <summary>
            /// n個の要素をparseして、それぞれにoffsetをaddした配列を返す
            /// </summary>
            /// <typeparam name="T"></typeparam>
            /// <param name="n"></param>
            /// <param name="offset"></param>
            /// <param name="parse"></param>
            /// <param name="add"></param>
            /// <returns></returns>
            T[] NextT<T>(int n, T offset, Func<string, T> parse, Func<T, T, T> add)
            {
                if (!accumulate(n)) return null;

                var a = new T[n];

                for (int i = 0; i < n; i++)
                    a[i] = add(parse(q.Dequeue()), offset);

                return a;
            }

            public string[] Next(int n) => NextT(n, "", x => x, (x, y) => x);
            public int[] NextInt(int n, int offset = 0) => NextT(n, offset, int.Parse, (x, y) => x + y);
            public long[] NextLong(int n, long offset = 0) => NextT(n, offset, long.Parse, (x, y) => x + y);
            public double[] NextDouble(int n, double offset = 0.0) => NextT(n, offset, double.Parse, (x, y) => x + y);
        }

        static class Utils
        {
            public static void Print<T>(IEnumerable<T> ie, string sep = " ") => Console.WriteLine(string.Join(sep, ie));
            public static void Print<T>(params T[] objs) => Print(objs);
            public static T Max<T>(params T[] objs) => objs.Max();
            public static T Min<T>(params T[] objs) => objs.Min();

            public static void Swap<T>(ref T x, ref T y) where T : struct { var t = x; x = y; y = t; }
            public static void Swap<T>(IList<T> list, int i, int j) where T : struct { var t = list[i]; list[i] = list[j]; list[j] = t; }

            /// <summary>
            /// vでfillされたT[d1][d2]配列を作成する
            /// </summary>
            /// <typeparam name="T"></typeparam>
            /// <param name="d1"></param>
            /// <param name="d2"></param>
            /// <param name="v"></param>
            /// <returns></returns>
            public static T[][] Create2DArray<T>(int d1, int d2, T v = default(T))
            {
                return
                    Enumerable.Repeat(0, d1).Select(_ =>
                    Enumerable.Repeat(v, d2).ToArray()).ToArray();
            }

            /// <summary>
            /// vでfillされたT[d1][d2][d3]配列を作成する
            /// </summary>
            /// <typeparam name="T"></typeparam>
            /// <param name="d1"></param>
            /// <param name="d2"></param>
            /// <param name="d3"></param>
            /// <param name="v"></param>
            /// <returns></returns>
            public static T[][][] Create3DArray<T>(int d1, int d2, int d3, T v = default(T))
            {
                return
                    Enumerable.Repeat(0, d1).Select(_ =>
                    Enumerable.Repeat(0, d2).Select(__ =>
                     Enumerable.Repeat(v, d3).ToArray()).ToArray()).ToArray();
            }
        }
        #endregion
    }
}