Problem:
Score : 800 points

Problem Statement

You are given a polynomial of degree N with integer coefficients: f(x)=a_Nx^N+a_{N-1}x^{N-1}+...+a_0. Find all prime numbers p that divide f(x) for every integer x.

Constraints

0 \leq N \leq 10^4

|a_i| \leq 10^9(0\leq i\leq N)

a_N \neq 0

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_N
:
a_0

Output

Print all prime numbers p that divide f(x) for every integer x, in ascending order.

Sample Input 1

2
7
-7
14

Sample Output 1

2
7

2 and 7 divide, for example, f(1)=14 and f(2)=28.

Sample Input 2

3
1
4
1
5

Sample Output 2

There may be no integers that satisfy the condition.

Sample Input 3

0
998244353

Sample Output 3

998244353

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

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

partial class Solver
{
    public class PrimeSet : List<int>
    {
        private bool[] isPrime;

        public PrimeSet(int size)
            : base()
        {
            isPrime = Enumerable.Repeat(true, size).ToArray();
            Add(2);
            for (int i = 3; i <= size; i += 2)
            {
                if (!isPrime[i >> 1]) continue;
                Add(i);
                for (long j = (long)i * i; j <= size; j += i + i)
                    isPrime[j >> 1] = false;
            }
        }

        public bool IsPrime(long x)
        {
            if (x == 2) return true;
            if (x <= 1 || x % 2 == 0) return false;
            return isPrime[x >> 1];
        }
    }


    static public List<long> Factor(PrimeSet primes, long x)
    {
        var result = new List<long>();

        foreach (var p in primes)
        {
            if (x % p == 0)
            {
                result.Add(p);

                while (x % p == 0) x /= p;
            }
        }

        if (x > 1) result.Add(x);

        return result;
    }

    public long ModFunction(long[] A, int x, long mod)
    {
        var c = 1L;
        var result = 0L;

        for (int i = 0; i < A.Length; i++)
        {
            result += c * (A[i] % mod);
            result %= mod;

            c = c * x % mod;
        }

        return result;
    }

    public void Run()
    {
        var primes = new PrimeSet((int)Math.Sqrt(1000 * 1000000000L) + 10);

        var N = ni();
        var A = nl(N + 1);
        Array.Reverse(A);

        var a0 = A[0];
        var sum = A.Sum();

        var gcd = (long)BigInteger.GreatestCommonDivisor(Math.Abs(a0), Math.Abs(sum));

        var candidate = Factor(primes, gcd);

        for (int i = -10; i < 10; i++)
        {
            candidate = candidate.Where(c => ModFunction(A, i, c) == 0).ToList();
        }

        foreach (var c in candidate)
        {
            cout.WriteLine(c);
        }
    }
}

// PREWRITEN CODE BEGINS FROM HERE
partial class Solver : Scanner
{
    public static void Main(string[] args)
    {
        new Solver(Console.In, Console.Out).Run();
    }

    private readonly TextReader cin;
    private readonly TextWriter cout;

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

    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 string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
}

public class Scanner
{
    private TextReader Reader;
    private Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

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

    public Scanner(TextReader reader)
    {
        this.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 (int i = 0; i < size; i++) array[i] = Next();
        return array;
    }
    public int[] NextIntArray(int size)
    {
        var array = new int[size];
        for (int i = 0; i < size; i++) array[i] = NextInt();
        return array;
    }

    public long[] NextLongArray(int size)
    {
        var array = new long[size];
        for (int i = 0; i < size; i++) array[i] = NextLong();
        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 bool StockTokens()
    {
        while (true)
        {
            var line = Reader.ReadLine();
            if (line == null) return false;
            var tokens = line.Trim().Split(" ".ToCharArray(), StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) continue;
            foreach (var token in tokens)
                TokenQueue.Enqueue(token);
            return true;
        }
    }
}