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 <bits/stdc++.h>
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define rep(i,N) for(long long i = 0; i < (long long)(N); i++)
#define repr(i,N) for(long long i = (long long)(N) - 1; i >= 0; i--)
#define rep1(i,N) for(long long i = 1; i <= (long long)(N) ; i++)
#define repr1(i,N) for(long long i = (N) ; (long long)(i) > 0 ; i--)
#define each(x,v) for(auto& x : v)
#define all(v) (v).begin(),(v).end()
#define sz(v) ((int)(v).size())
#define ini(...) int __VA_ARGS__; in(__VA_ARGS__)
#define inl(...) long long __VA_ARGS__; in(__VA_ARGS__)
#define ins(...) string __VA_ARGS__; in(__VA_ARGS__)
using namespace std; void solve();
using ll = long long; template<class T = ll> using V = vector<T>;
using vi = V<int>; using vl = V<>; using vvi = V< V<int> >;
constexpr int inf = 1001001001; constexpr ll infLL = (1LL << 61) - 1;
struct IoSetupNya {IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7);} } iosetupnya;
template<typename T, typename U> inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; }
template<typename T, typename U> inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; }
template<typename T, typename U> ostream& operator <<(ostream& os, const pair<T, U> &p) { os << p.first << " " << p.second; return os; }
template<typename T, typename U> istream& operator >>(istream& is, pair<T, U> &p) { is >> p.first >> p.second; return is; }
template<typename T> ostream& operator <<(ostream& os, const vector<T> &v) { int s = (int)v.size(); rep(i,s) os << (i ? " " : "") << v[i]; return os; }
template<typename T> istream& operator >>(istream& is, vector<T> &v) { for(auto &x : v) is >> x; return is; }
void in(){} template <typename T,class... U> void in(T &t,U &...u){ cin >> t; in(u...);}
void out(){cout << "\n";} template <typename T,class... U> void out(const T &t,const U &...u){ cout << t; if(sizeof...(u)) cout << " "; out(u...);}
template<typename T>void die(T x){out(x); exit(0);}
#ifdef NyaanDebug
  #include "NyaanDebug.h"
  #define trc(...) do { cerr << #__VA_ARGS__ << " = "; dbg_out(__VA_ARGS__);} while(0)
  #define trca(v,N) do { cerr << #v << " = "; array_out(v , N);cout << endl;} while(0)
#else
  #define trc(...)
  #define trca(...)
  int main(){solve();}
#endif
 
#define inc(...) char __VA_ARGS__; in(__VA_ARGS__)
#define in2(N,s,t) rep(i,N){in(s[i] , t[i]);}
#define in3(N,s,t,u) rep(i,N){in(s[i] , t[i] , u[i]);}
using vd = V<double>; using vs = V<string>; using vvl = V< V<> >;
template<typename T>using heap = priority_queue< T , V<T> , greater<T> >;
 
using P = pair<ll,ll>; using vp = V<P>;
constexpr int MOD = /**/ 1000000007; //*/ 998244353;
//////////////////


 
// 素数判定 O( sqrt(N) log log N )
// 0からNに対して素数->1、それ以外->0の配列を返す関数
vector<int> Primes(int N){
  vector<int> A(N + 1 , 1);
  A[0] = A[1] = 0;
  for(int i = 2; i * i <= N ; i++) 
    if(A[i]==1) for(int j = i << 1 ; j <= N; j += i) A[j] = 0;
  return A;
}
 
// 因数 O( sqrt(N) log log N )
// 0からNに対して素数->1、それ以外->最小の素数である因数、の配列を返す
vector<int> Factors(int N){
  vector<int> A(N + 1 , 1);
  A[0] = A[1] = 0;
  for(int i = 2; i * i <= N ; i++) 
    if(A[i]==1) for(int j = i << 1 ; j <= N; j += i) A[j] = i;
  return A;
}
 
// オイラーのトーシェント関数 φ(N)=(Nと互いに素なN以下の自然数の個数)
vector<int> EulersTotientFunction(int N){
  vector<int> ret(N + 1 , 0);
  for(int i = 0; i <= N ; i++) ret[i] = i; 
  for(int i = 2 ; i <= N ; i++){
    if(ret[i] == i)
      for(int j = i; j <= N; j += i) ret[j] = ret[j] / i * (i - 1);
  }
  return ret;
}
 
// 約数列挙 O(sqrt(N))
// Nの約数を列挙した配列を返す
vector<long long> Divisor(long long N){
  vector<long long> v;
  for(long long i = 1; i * i <= N ; i++){
    if(N % i == 0){
      v.push_back(i);
      if(i * i != N) v.push_back(N / i);
    }
  }
  return v;
}
 
// 素因数分解
// 因数をkey、そのべきをvalueとするmapを返す
// ex) N=12 -> m={ (2,2) , (3,1) }
map<long long,int> PrimeFactors(long long N){
  map<long long,int> m;
  for(long long i=2; i * i <= N; i++)
    while(N % i == 0) m[i]++ , N /= i;
  if(N != 1) m[N]++;
  return m;
}
 
// 原始根 modでrが原始根かどうかを調べる
bool PrimitiveRoot(long long r , long long mod){
  r %= mod; if(r == 0) return false;
  auto modpow = [](long long a,long long b,long long m)->long long{
    a %= m; long long ret = 1;
    while(b){
      if(b & 1) ret = a * ret % m;
      a = a * a % m;
      b >>= 1;
    }
    return ret;
  };
  map<long long,int> m = PrimeFactors(mod - 1);
  each(x , m){
    if(modpow(r , (mod - 1) / x.fi , mod ) == 1) return false;
  }
  return true;
}
 
// 拡張ユークリッド ax+by=gcd(a,b)の解
// 返り値　最大公約数
long long extgcd(long long a,long long b, long long &x, long long &y){
  if(b == 0){
    x = 1; y = 0; return a;
  }
  long long d = extgcd(b , a%b , y , x);
  y -= a / b * x;
  return d;
}
 
// c++17での名前衝突を避けるためdefine
#define gcd nyagcd
#define lcm nyalcm
ll nyagcd(ll x, ll y){
  ll z;
  if(x > y) swap(x,y);
  while(x){
    x = y % (z = x); y = z;
  }    
  return y;
}
ll nyalcm(ll x,ll y){
  return 1LL * x / gcd(x,y) * y;
}
 
void solve(){
  ini(N);
  vl a(N + 1); in(a);
  reverse(all(a));
  ll g = 0;
  each(x , a) g = gcd(g , abs(x));
  if(g == 0) {exit(1);}
  each(x , a) x /= g;

  // a / b
  auto div = [](vl a,vl b,ll mod) -> vl{
    reverse(all(a)); reverse(all(b));
    each(x , a) x = (x % mod + mod) % mod;
    each(x , b) x = (x % mod + mod) % mod;
    trc(a); trc(b);
    for(int i = 0 ; i + sz(b) <= sz(a) ; i++){
      if(a[i] == 0) continue;
      ll c = a[i];
      for(int j = 0 ; j < sz(b) ; j++){
        a[i + j] -= c * b[j] % mod;
        if(a[i + j] < 0) a[i + j] += mod;
      }
    }
    reverse(all(a));
    return a;
  };
  
  vl nya; nya.reserve(16384);
  auto f = [&a , &div , &nya](ll n) -> bool{
    trc(n);
    if((int)a.size() + 10 < n) return false; 
    nya.resize(n + 1);
    fill(all(nya) , 0);
    nya[n] = 1 , nya[1] = n - 1;
    trc(nya);
    /*
    auto d = div(a , nya , n);
    trc(d);
    each(x , d) if(x != 0) return false;
    */
    return true;
  };
 
  vl ans;
  auto gfac = PrimeFactors(g);
  each(x , gfac) ans.pb(x.first);
  ll a0 = a[0];
  auto fac = PrimeFactors(a0);
  each(x , fac){
    trc(x);
    if(f(x.first)) ans.pb(x.first);
  }
  sort(all(ans));
  ans.erase(unique(all(ans)) , ans.end());
  each(x , ans) out(x);
}