#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using Int = long long; template ostream &operator<<(ostream &os, const pair &a) { return os << "(" << a.first << ", " << a.second << ")"; }; template void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; } template bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; } template bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; } int root(vector &uf, int u) { return (uf[u] < 0) ? u : (uf[u] = root(uf, uf[u])); } bool connect(vector &uf, int u, int v) { u = root(uf, u); v = root(uf, v); if (u == v) return false; if (uf[u] > uf[v]) swap(u, v); uf[u] += uf[v]; uf[v] = u; return true; } int M; vector oks; map, int> cache; int calc(vector fs) { vector uf(M, -1); for (int a = 1; a < M; ++a) if (fs[a]) { for (int u = 0; u < M; ++u) { connect(uf, u, (Int)u * a % M); } } for (int a = 1; a < M; ++a) if (root(uf, 1) == root(uf, a)) { fs[a] = true; } auto it = cache.find(fs); if (it != cache.end()) { return it->second; } set app; set used; for (int a = 1; a < M; ++a) if (oks[a] && !fs[a]) { if (used.insert(root(uf, a)).second) { auto gs = fs; gs[a] = true; const int res = calc(gs); app.insert(res); } } int ret = -1; for (int g = 0; ; ++g) { if (!app.count(g)) { ret = g; break; } } return cache[fs] = ret; } int brute(int m) { M = m; oks.assign(m, false); for (int a = 1; a < m; ++a) { oks[a] = (__gcd(m, a) == 1); } cache.clear(); vector start(m, false); return calc(start); } constexpr int LIM = 100'010; int lpf[LIM]; map> mg(int m) { map> pess; for (; m > 1; ) { const int p = lpf[m]; int e = 0; do { ++e; m /= p; } while (m % p == 0); if (p == 2) { if (e == 1) { // } else if (e == 2) { pess[2].push_back(1); } else { pess[2].push_back(1); pess[2].push_back(e - 2); } } else { for (int n = p - 1; n > 1; ) { const int q = lpf[n]; int f = 0; do { ++f; n /= q; } while (n % q == 0); pess[q].push_back(f); } if (e >= 2) { pess[p].push_back(e - 1); } } } for (auto &kv : pess) { sort(kv.second.begin(), kv.second.end()); } return pess; } int solve(int m) { if(m<=1000)return brute(m); const auto pess = mg(m); int ret = 0; for (const auto &kv : pess) { for (const int e : kv.second) { ret += e; } } return ret; } int main() { iota(lpf, lpf + LIM, 0); for (int p = 2; p < LIM; ++p) if (lpf[p] == p) { for (int n = 2 * p; n < LIM; n += p) chmin(lpf[n], p); } /* for (int m = 1; m <= 100; ++m) { const int brt = brute(m); const int slv = solve(m); const auto pess = mg(m); if (brt != slv) { cerr << m << ": " << brt << " " << slv << "; "; for (const auto &kv : pess) { cerr << " " << kv.first << ":"; for (const int e : kv.second) cerr << e; } cerr << endl; } } cout << endl; //*/ /* 8: 0 2; 2:11 12: 0 2; 2:11 24: 1 3; 2:111 40: 2 4; 2:112 48: 2 4; 2:112 56: 2 4; 2:111 3:1 60: 2 4; 2:112 63: 0 4; 2:11 3:11 65: 1 5; 2:22 3:1 72: 2 4; 2:111 3:1 80: 1 5; 2:122 84: 2 4; 2:111 3:1 88: 2 4; 2:111 5:1 91: 1 5; 2:12 3:11 */ int N; for (; ~scanf("%d", &N); ) { vector A(N); for (int i = 0; i < N; ++i) { scanf("%d", &A[i]); } sort(A.begin(), A.end()); int ans = 0; for (int i = 0, j; i < N; i = j) { for (j = i; j < N && A[i] == A[j]; ++j) {} if ((j - i) % 2 != 0) { ans ^= solve(A[i]); } } puts(ans ? "Y" : "X"); } return 0; }