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++14 (GCC 5.4.1) (judge verdict: Runtime Error)

#include <iostream>
#include <algorithm>
#include <vector>
#include <string.h>
#include <map>
#include <random>
using namespace std;
using lint = long long;
const lint mod = 998244353;
#define all(x) (x).begin(), (x).end()
#define bitcount(n) __builtin_popcountl((lint)(n))
#define fcout cout << fixed << setprecision(15)
#define highest(x) (63 - __builtin_clzl(x))
template<class T> inline void YES(T condition){ if(condition) cout << "YES" << endl; else cout << "NO" << endl; }
template<class T> inline void Yes(T condition){ if(condition) cout << "Yes" << endl; else cout << "No" << endl; }
template<class T = string, class U = char>int character_count(T text, U character){ int ans = 0; for(U i: text){ ans += (i == character); } return ans; }
lint power(lint base, lint exponent, lint module){ if(exponent % 2){ return power(base, exponent - 1, module) * base % module; }else if(exponent){ lint root_ans = power(base, exponent / 2, module); return root_ans * root_ans % module; }else{ return 1; }}
struct position{ int y, x; }; position mv[4] = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}}; // double euclidean(position first, position second){ return sqrt((second.x - first.x) * (second.x - first.x) + (second.y - first.y) * (second.y - first.y)); }
template<class T, class U> string to_string(pair<T, U> x){ return to_string(x.first) + "," + to_string(x.second); } string to_string(string x){ return x; }
template<class itr> void array_output(itr start, itr goal){ string ans; for(auto i = start; i != goal; i++) ans += to_string(*i) + " "; if(!ans.empty()) ans.pop_back(); cout << ans << endl; }
template<class itr> void cins(itr first, itr last){ for(auto i = first; i != last; i++){ cin >> (*i); } }
template<class T> T gcd(T a, T b){ if(a && b){ return gcd(min(a, b), max(a, b) % min(a, b)); }else{ return a; }} template<class T> T lcm(T a, T b){ return a / gcd(a, b) * b; }
struct combination{ vector<lint> fact, inv; combination(int sz) : fact(sz + 1), inv(sz + 1){ fact[0] = 1; for(int i = 1; i <= sz; i++){ fact[i] = fact[i - 1] * i % mod; } inv[sz] = power(fact[sz], mod - 2, mod); for(int i = sz - 1; i >= 0; i--){ inv[i] = inv[i + 1] * (i + 1) % mod; } } lint C(int p, int q) const{ if(q < 0 || p < q) return 0; return (fact[p] * inv[q] % mod * inv[p - q] % mod); } };
template<class itr> bool next_sequence(itr first, itr last, int max_bound){ itr now = last; while(now != first){ now--; (*now)++; if((*now) == max_bound){ (*now) = 0; }else{ return true; } } return false; }

using int64 = long long;

map< int64, int > prime_factor(int64 n)
{
    map< int64, int > ret;
    for(int64 i = 2; i * i <= n; i++){
        while(n % i == 0) {
            ret[i]++;
            n /= i;
        }
    }
    if(n != 1) ret[n] = 1;
    return(ret);
}

random_device rnd;

int main(){
    int N;
    cin >> N;
    lint a[N + 1];
    for(int i = N; i >= 0; i--){
        cin >> a[i];
    }
    auto factor = prime_factor(abs(a[0]));
    for(auto i: factor){
        auto p = i.first;
        for(int j = 0; j < 10; j++){
            lint rd = rnd() % (p - 2) + 1;
            lint sum = 0;
            for(int k = 0; k <= N; k++){
                sum += ((a[k] + p * lint(1e9)) % p * power(rd, k, p) % p);
            }
            if(sum % p != 0){
                goto pass;
            }
        }
        cout << p << endl;
    pass:;
    }
}