Problem:
Score : 500 points

Problem Statement

We have a sequence of N integers: A_1, A_2, \cdots, A_N.

You can perform the following operation between 0 and K times (inclusive):

Choose two integers i and j such that i \neq j, each between 1 and N (inclusive). Add 1 to A_i and -1 to A_j, possibly producing a negative element.

Compute the maximum possible positive integer that divides every element of A after the operations. Here a positive integer x divides an integer y if and only if there exists an integer z such that y = xz.

Constraints

2 \leq N \leq 500

1 \leq A_i \leq 10^6

0 \leq K \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_{N-1} A_{N}

Output

Print the maximum possible positive integer that divides every element of A after the operations.

Sample Input 1

2 3
8 20

Sample Output 1

7

7 will divide every element of A if, for example, we perform the following operation:

Choose i = 2, j = 1. A becomes (7, 21).

We cannot reach the situation where 8 or greater integer divides every element of A.

Sample Input 2

2 10
3 5

Sample Output 2

8

Consider performing the following five operations:

Choose i = 2, j = 1. A becomes (2, 6).

Choose i = 2, j = 1. A becomes (1, 7).

Choose i = 2, j = 1. A becomes (0, 8).

Choose i = 2, j = 1. A becomes (-1, 9).

Choose i = 1, j = 2. A becomes (0, 8).

Then, 0 = 8 \times 0 and 8 = 8 \times 1, so 8 divides every element of A. We cannot reach the situation where 9 or greater integer divides every element of A.

Sample Input 3

4 5
10 1 2 22

Sample Output 3

7

Sample Input 4

8 7
1 7 5 6 8 2 6 5

Sample Output 4

5

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

using Training20200527.Questions;
using Training20200527.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace Training20200527.Questions
{
    /// <summary>
    /// https://atcoder.jp/contests/abc136/tasks/abc136_e
    /// </summary>
    public class QuestionC : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var nk = inputStream.ReadIntArray();
            var n = nk[0];
            var k = nk[1];
            var a = inputStream.ReadIntArray();
            var sum = a.Sum();
            var divisiors = GetDivisiors(sum);

            var max = 1;
            foreach (var divisior in divisiors)
            {
                if (CountToDivisibleBy(divisior, a) <= k)
                {
                    max = Math.Max(max, divisior);
                }
            }

            yield return max;
        }

        IEnumerable<int> GetDivisiors(int n)
        {
            for (int i = 1; i * i <= n; i++)
            {
                if (n % i == 0)
                {
                    var other = n / i;
                    yield return i;
                    if (i != other)
                    {
                        yield return other;
                    }
                }
            }
        }

        int CountToDivisibleBy(int divisior, int[] a)
        {
            var mods = a.Select(i => i % divisior).ToArray();
            Array.Sort(mods);

            var count = 0;
            var smallCursor = 0;
            var largeCursor = a.Length - 1;
            while (smallCursor < largeCursor)
            {
                var small = mods[smallCursor];
                var large = divisior - mods[largeCursor];
                if (small < large)
                {
                    count += small;
                    mods[largeCursor] += small;
                    smallCursor++;
                }
                else if (small > large)
                {
                    count += large;
                    mods[smallCursor] -= large;
                    largeCursor--;
                }
                else
                {
                    count += small;
                    smallCursor++;
                    largeCursor--;
                }
            }
            return count;
        }
    }
}


namespace Training20200527
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionC();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace Training20200527.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace Training20200527.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion