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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#define _CRT_SECURE_NO_WARNINGS
#include "bits/stdc++.h"
#if defined(_MSC_VER) || defined(ONLINE_JUDGE)
#define getchar_unlocked _getchar_nolock
#define putchar_unlocked _putchar_nolock
#endif
#define rep(i, n) for(int i=0; i<(n); ++i)
#define FOR(i, m, n) for(int i=(m); i<(n); ++i)
#define sz(x) ((int)(x).size())
#define all(x) (x).begin(),(x).end()
#define rall(x) (x).rbegin(),(x).rend()
#define mp make_pair
#define pb push_back
#define dump(x) cerr << #x << " = " << (x) << endl;

using namespace std;
using LL=long long;
using VI=vector<int>;      using VL=vector<LL>;
using VS=vector<string>;   using VD=vector<double>;
using VVI=vector<VI>;      using VVL=vector<VL>;
using PII=pair<int, int>;  using PLL=pair<LL, LL>;
const int inf = (int)1e9;  const double pi = acos(-1.0);
const LL MOD = 1000000007; const string Snum = "0123456789";
const int dx[] = { -1,0,1,0,-1,-1,1,1 }, dy[] = { 0,1,0,-1,-1,1,-1,1 };

template<class T>void Sort(T& a) { sort(all(a)); }
template<class T>void RSort(T& a) { sort(rall(a)); }
template<class T>void Reverse(T& a) { reverse(all(a)); }
template<class T>void Unique(T& a) { a.erase(unique(all(a)), a.end()); }
template<class T>auto Max(const T& a) { return *max_element(all(a)); }
template<class T>auto Min(const T& a) { return *min_element(all(a)); }
template<class T>int MaxPos(const T& a) { return max_element(all(a)) - a.begin(); }
template<class T>int MinPos(const T& a) { return min_element(all(a)) - a.begin(); }
template<class T, class U>int Count(const T& a, U v) { return count(all(a), v); }
template<class T, class U>int Find(const T& a, U v) { auto pos = find(all(a), v); return pos == a.end() ? -1 : pos - a.begin(); }
template<class T, class U>U Sum(const T& a, U v) { return accumulate(all(a), v); }
template<class T, class U>int Lower(const T& a, U v) { return lower_bound(all(a), v) - a.begin(); }
template<class T, class U>int Upper(const T& a, U v) { return upper_bound(all(a), v) - a.begin(); }
template<class T, class P>int CountIf(const T& a, P f) { return count_if(all(a), f); }
template<class T, class P>void RemoveIf(T& a, P f) { a.erase(remove_if(all(a), f), a.end()); }
template<class T>T Age(T n, T m) { return (n + m - 1) / m; }
template<class T>T Gcd(T n, T m) { return m == 0 ? n : Gcd(m, n % m); }
template<class T>T Lcm(T n, T m) { return n / Gcd(n, m) * m; }
template<class T>T Pow(T a, T n) { T r = 1; while (n > 0) { if (n & 1)r *= a; a *= a; n /= 2; }return r; }
template<class T>T Powmod(T a, T n, T m = MOD) { T r = 1; while (n > 0) { if (n & 1)r = r * a % m, n--; else a = a * a % m, n /= 2; }return r; }
string operator*(string s, int n) { string ret; rep(i, n)ret += s; return ret; }

/* INPUT */
template<class T>T InputF() { T ret; cin >> ret; return ret; }
template<> char InputF() {
	char ret = '\0', c = getchar_unlocked();
	while (c != '\0' && c != '\n' && c != '\t' && c != ' ') { ret = c; c = getchar_unlocked(); }
	return ret;
}
template<> string InputF() {
	string ret; char c = getchar_unlocked();
	while (c != '\0' && c != '\n' && c != '\t' && c != ' ') { ret.push_back(c); c = getchar_unlocked(); }
	return ret;
}
template<> int InputF() {
	int ret = 0; bool neg = false; char c = getchar_unlocked();
	if (c == '-') { neg = true; c = getchar_unlocked(); }
	while ('0' <= c && c <= '9') { ret = ret * 10 + (c - '0'); c = getchar_unlocked(); }
	return neg ? -ret : ret;
}
template<> LL InputF() {
	LL ret = 0; bool neg = false; char c = getchar_unlocked();
	if (c == '-') { neg = true; c = getchar_unlocked(); }
	while ('0' <= c && c <= '9') { ret = ret * 10LL + (LL(c) - LL('0')); c = getchar_unlocked(); }
	return neg ? -ret : ret;
}
template<> double InputF() {
	double ret = 0, dp = 1; bool neg = false, adp = false; char c = getchar_unlocked();
	if (c == '-') { neg = true; c = getchar_unlocked(); }
	while (('0' <= c && c <= '9') || c == '.') {
		if (c == '.') { adp = true; }
		else if (adp) { dp *= 0.1; ret += (c - '0') * dp; }
		else { ret = ret * 10.0 + (c - '0'); }
		c = getchar_unlocked();
	}
	return neg ? -ret : ret;
}
string GetLine() {
	string ret; char c = getchar_unlocked();
	while (c != '\0' && c != '\n') { ret.push_back(c); c = getchar_unlocked(); }
	return ret;
}

template<class T>T StoT(string s) { return s; }
template<> char StoT(string s) { return s.front(); }
template<> string StoT(string s) { return s; }
template<> int StoT(string s) { return stoi(s); }
template<> LL StoT(string s) { return stoll(s); }
template<> double StoT(string s) { return stod(s); }

VS InputB;
template<class T>struct InputC {
	operator T() { return InputF<T>(); }
	vector<T> operator[](int h) {
		vector<T> ret(h);
		rep(i, h)ret[i] = InputF<T>();
		return ret;
	}
	vector<vector<T>> operator[](PII p) {
		int h, w; tie(h, w) = p;
		vector<vector<T>> ret(h, vector<T>(w));
		rep(i, h)rep(j, w)ret[i][j] = InputF<T>();
		return ret;
	}
	T operator()(int n) {
		while (sz(InputB) <= n)InputB.pb(InputF<string>());
		return StoT<T>(InputB[n]);
	}
	vector<T> operator()(int n, int h) {
		vector<T> ret(h);
		rep(i, h)ret[i] = InputC::operator()(i + n);
		return ret;
	}
	vector<vector<T>> operator()(int n, int h, int w) {
		vector<vector<T>> ret(h, vector<T>(w));
		rep(i, h)rep(j, w)ret[i][j] = InputC::operator()(n + i * w + j);
		return ret;
	}
};
InputC<char> inc; InputC<string> ins; InputC<int> ini; InputC<LL> inl; InputC<double> ind;

/* OUTPUT */
class ostreamfbuff : public basic_streambuf<char, char_traits<char>> {
protected:int overflow(int nCh = EOF) { putchar_unlocked(nCh); return 0; }
};
class ostreamf : public basic_iostream<char, char_traits<char>> {
public:ostreamf(void) : basic_iostream<char, char_traits<char>>(new ostreamfbuff()) {}
};
ostreamf coutf; ostream& operator<<(ostream& ostr, InputC<string> i) { ostr << string(i); return ostr; }
struct BoolStr { string t, f; BoolStr(string _t, string _f) :t(_t), f(_f) {} }yes("yes", "no"), Yes("Yes", "No"), YES("YES", "NO"), Yay("Yay!", ":(");
struct Divizer { string s; Divizer(string _s) :s(_s) {} }spc(" "), nosp(""), comma(","), lin("\n");
struct Output {
	bool flag = false; BoolStr bs; Divizer di;
	Output(bool f, BoolStr b, Divizer d) :flag(f), bs(b), di(d) {}
	template<class T>void print(T o) { coutf << o; }
	void print(double o) { printf("%.20f", o); }
	void print(bool o) { coutf << (o ? bs.t : bs.f); }
	template<class T, class U>void print(const pair<T, U>& o) {
		print(o.first); coutf << di.s; print(o.second);
	}
	template<class T>void print(const vector<T>& o) {
		for (int i = 0; i < (int)o.size(); ++i) { if (i != 0)coutf << di.s; print(o[i]); }
	}
	template<class T>void print(const vector<vector<T>>& o) {
		for (int i = 0; i < (int)o.size(); ++i) { if (i != 0)coutf << '\n'; print(o[i]); }
	}
	template<class T, class U>void print(const map<T, U>& o) {
		coutf << "Map"; for (auto a : o) { coutf << " {"; print(a); coutf << "},"; } coutf << '\n';
	}
	template<class T>void print(const set<T>& o) {
		coutf << "Set {"; for (auto a : o) { print(a); coutf << ","; } coutf << "}\n";
	}
	template<class T>Output operator ,(T o) {
		if (flag)coutf << di.s; print(o); Output t(true, bs, di); return t;
	}
	Output operator,(Output o) { coutf << '\n'; Output t(false, bs, di); return t; }
	Output operator,(BoolStr b) { Output t(flag, b, di); return t; }
	Output operator,(Divizer d) { Output t(flag, bs, d); return t; }
}out(false, Yes, spc);

/* ANSWER */
struct Answer {
	int mini = INT_MAX, maxi = INT_MIN, sumi = 0;
	LL minl = LLONG_MAX, maxl = LLONG_MIN, suml = 0;
	double mind = DBL_MAX, maxd = DBL_MIN, sumd = 0; int cnt = 0;
	void operator=(int n) { mini = min(mini, n); maxi = max(maxi, n); sumi += n; cnt++; }
	void operator=(LL n) { minl = min(minl, n); maxl = max(maxl, n); suml += n; cnt++; }
	void operator=(double n) { mind = min(mind, n); maxd = max(maxd, n); sumd += n; cnt++; }
}ans;


template<class T>vector<T> DivAll(T n) {
	vector<T> ret; T i = 1;
	for (; i * i < n; ++i)if (n % i == 0)ret.push_back(i);
	bool flag = i * i == n; if (flag)ret.push_back(i);
	for (i = (int)ret.size() - 1 - flag; i >= 0; --i)ret.push_back(n / ret[i]);
	return ret;
}

int main() {
	LL n = inl, k = inl;
	VL a = inl[n];

	LL sum = Sum(a, 0LL);
	auto div = DivAll(sum);
	Reverse(div);

	for (auto D : div) {
		vector<PLL> ud;
		rep(i, n) {
			LL Up = Age(a[i], D) * D - a[i];
			LL Down = a[i] - a[i] / D * D;

			if (Up != 0 && Down != 0) {
				ud.pb(mp(Up, Down));
			}
		}

		Sort(ud);

		bool flag = false;

		if (ud.empty()) {
			flag = true;
		}
		else {
			{
				LL sum = ud[0].first, sum2 = 0;
				FOR(i, 1, sz(ud)) {
					if (sum > 0) { sum -= ud[i].second; sum2 += ud[i].second; }
					else { sum += ud[i].first; }
				}
				if (sum == 0 && sum2 <= k)flag = true;
			}
			{
				LL sum = ud[0].second, sum2 = 0;
				FOR(i, 1, sz(ud)) {
					if (sum > 0) { sum -= ud[i].second; sum2 += ud[i].second; }
					else { sum += ud[i].first; }
				}
				if (sum == 0 && sum2 <= k)flag = true;
			}
		}

		if (flag) {
			out, D, out;
			break;
		}
	}
}