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: Wrong Answer)

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
using System.Numerics;
using static System.Math;
using Number = System.Int64;
namespace Program {
    public class Solver {
        Random rnd = new Random(0);

        public void Solve() {
            var n = ri;
            var a = new List<long>();
            for (int i = 0; i <= n; i++)
                a.Add(rl);
            while (a.Last() == 0) a.RemoveAt(a.Count - 1);
            Debug.WriteLine(a.AsJoinedString());
            var fac = new SortedSet<int>();
            var N = (int)Abs(a.Last());
            for (int i = 2; i * i <= N; i++) {
                if (N % i != 0) continue;
                fac.Add(i);
                while (N % i == 0) N /= i;
            }
            Debug.WriteLine(N);
            if (N != 1) fac.Add(N);
            foreach (var x in MathEx.SieveList(n + 5)) fac.Add(x);
            foreach (var p in fac) {
                var ok = true;
                for (int k = 0; k < 200; k++) {
                    long x = rnd.Next(1, 1000000000);
                    var v = 0L;
                    foreach (var c in a) v = (v * x + c) % p;
                    ok &= v == 0;
                    if (!ok) break;
                }
                if (ok) Console.WriteLine(p);
            }
        }



        const long INF = 1L << 60;
        int ri => sc.Integer();
        long rl => sc.Long();
        double rd => sc.Double();
        string rs => sc.Scan();
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static IEnumerable<int> Rep(int n) => Enumerable.Range(0, n);
        static IEnumerable<int> RRep(int n) => Enumerable.Range(0, n).Reverse();
        static T[] Enumerate<T>(int n, Func<int, T> f) {
            var a = new T[n];
            for (int i = 0; i < a.Length; ++i) a[i] = f(i);
            return a;
        }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex {
    static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
        return string.Join(st, ie);
    }
    static public void Main() {
        Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
        var solver = new Program.Solver();
        solver.Solve();
        Console.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO {
    using System.IO;
    using System.Text;
    using System.Globalization;

    public class Printer : StreamWriter {
        public override IFormatProvider FormatProvider => CultureInfo.InvariantCulture;
        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
    }

    public class StreamScanner {
        public StreamScanner(Stream stream) { str = stream; }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;

        private byte read() {
            if (isEof) return 0;
            if (ptr >= len) {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0) {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }

        public char Char() {
            byte b = 0;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }
        public string Scan() {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine() {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
        public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
        public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
}

#endregion

#region sieve O(NloglogN)
static public partial class MathEx {
    static public bool[] Sieve(int p, List<int> primes = null) {
        var isPrime = new bool[p + 1];
        for (int i = 2; i < isPrime.Length; i++) isPrime[i] = true;
        for (int i = 2; i * i <= p; i++)
            if (!isPrime[i]) continue;
            else for (int j = i * i; j <= p; j += i) isPrime[j] = false;
        if (primes != null) for (int i = 0; i <= p; i++) if (isPrime[i]) primes.Add(i);

        return isPrime;
    }
    static public List<int> SieveList(int p) { var ret = new List<int>(); Sieve(p, ret); return ret; }
}
#endregion