#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define int long long #define rep(i, n) for (int i = 0; i < (int)(n); i++) // [0, b) #define rep2(i, a, b) for (int i = (int)(a); i < (int)(b); i++) // [a, b) #define rep3(i, a, b) for (int i = (int)(a); i >= (int)(b); i--) // reversed [b, a] so [a, a-1, a-2, ..., b] #define FOR(i, a) for (auto &i: a) #define ALL(obj) begin(obj), end(obj) #define MAX(x) *max_element(ALL(x)) #define MIN(x) *min_element(ALL(x)) #define SUM(x) accumulate(ALL(x), 0LL) #define LOWER_BOUND(A, key) distance(A.begin(), lower_bound(ALL(A), key)) #define UPPER_BOUND(A, key) distance(A.begin(), upper_bound(ALL(A), key)) using namespace std; const int MOD = 998244353; const int MOD2 = 1000000007; const int INF = (int)(1e13 + 7); const double EPS = 1e-14; const double PI = acos(-1); int CEIL(int a, int b) { return (a >= 0 ? (a + (b - 1)) / b : (a - (b - 1)) / b); } //ceil() for int int mod(int a, int b) { return a >= 0 ? a % b : a - (b * CEIL(a, b)); } //always return positive num int pow_mod(int a, int b) { //return x^y in order(log(y)) int res = 1; for (a %= MOD; b; a = a * a % MOD, b >>= 1) if (b & 1) res = res * a % MOD; return res; } namespace FastPrimeFactorization { template struct UnsafeMod { UnsafeMod(): x(0) {} UnsafeMod(word _x): x(init(_x)) {} bool operator==(const UnsafeMod &rhs) const { return x == rhs.x; } bool operator!=(const UnsafeMod &rhs) const { return x != rhs.x; } UnsafeMod &operator+=(const UnsafeMod &rhs) { if ((x += rhs.x) >= mod) x -= mod; return *this; } UnsafeMod &operator-=(const UnsafeMod &rhs) { if (sword(x -= rhs.x) < 0) x += mod; return *this; } UnsafeMod &operator*=(const UnsafeMod &rhs) { x = reduce(dword(x) * rhs.x); return *this; } UnsafeMod operator+(const UnsafeMod &rhs) const { return UnsafeMod(*this) += rhs; } UnsafeMod operator-(const UnsafeMod &rhs) const { return UnsafeMod(*this) -= rhs; } UnsafeMod operator*(const UnsafeMod &rhs) const { return UnsafeMod(*this) *= rhs; } UnsafeMod pow(uint64_t e) const { UnsafeMod ret(1); for (UnsafeMod base = *this; e; e >>= 1, base *= base) { if (e & 1) ret *= base; } return ret; } word get() const { return reduce(x); } static constexpr int word_bits = sizeof(word) * 8; static word modulus() { return mod; } static word init(word w) { return reduce(dword(w) * r2); } static void set_mod(word m) { mod = m; inv = mul_inv(mod); r2 = -dword(mod) % mod; } static word reduce(dword x) { word y = word(x >> word_bits) - word((dword(word(x) * inv) * mod) >> word_bits); return sword(y) < 0 ? y + mod : y; } static word mul_inv(word n, int e = 6, word x = 1) { return !e ? x : mul_inv(n, e - 1, x * (2 - x * n)); } static word mod, inv, r2; word x; }; using uint128_t = __uint128_t; using Mod64 = UnsafeMod; template<> uint64_t Mod64::mod = 0; template<> uint64_t Mod64::inv = 0; template<> uint64_t Mod64::r2 = 0; using Mod32 = UnsafeMod; template<> uint32_t Mod32::mod = 0; template<> uint32_t Mod32::inv = 0; template<> uint32_t Mod32::r2 = 0; bool miller_rabin_primality_test_uint64(uint64_t n) { Mod64::set_mod(n); uint64_t d = n - 1; while (d % 2 == 0) d /= 2; Mod64 e{1}, rev{n - 1}; // http://miller-rabin.appspot.com/ < 2^64 for (uint64_t a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) { if (n <= a) break; uint64_t t = d; Mod64 y = Mod64(a).pow(t); while (t != n - 1 && y != e && y != rev) { y *= y; t *= 2; } if (y != rev && t % 2 == 0) return false; } return true; } bool miller_rabin_primality_test_uint32(uint32_t n) { Mod32::set_mod(n); uint32_t d = n - 1; while (d % 2 == 0) d /= 2; Mod32 e{1}, rev{n - 1}; for (uint32_t a: {2, 7, 61}) { if (n <= a) break; uint32_t t = d; Mod32 y = Mod32(a).pow(t); while (t != n - 1 && y != e && y != rev) { y *= y; t *= 2; } if (y != rev && t % 2 == 0) return false; } return true; } bool is_prime(uint64_t n) { if (n == 2) return true; if (n == 1 || n % 2 == 0) return false; if (n < uint64_t(1) << 31) return miller_rabin_primality_test_uint32(n); return miller_rabin_primality_test_uint64(n); } uint64_t pollard_rho(uint64_t n) { if (is_prime(n)) return n; if (n % 2 == 0) return 2; Mod64::set_mod(n); uint64_t d; Mod64 one{1}; for (Mod64 c{one};; c += one) { Mod64 x{2}, y{2}; do { x = x * x + c; y = y * y + c; y = y * y + c; d = __gcd((x - y).get(), n); } while (d == 1); if (d < n) return d; } assert(0); } vector prime_factor(uint64_t n) { if (n <= 1) return {}; uint64_t p = pollard_rho(n); if (p == n) return {p}; auto l = prime_factor(p); auto r = prime_factor(n / p); copy(begin(r), end(r), back_inserter(l)); return l; } }; // namespace FastPrimeFactorization signed main() { cin.tie(nullptr); cout.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); int n; cin >> n; rep(i, n) { int x; cin >> x; cout << x << " " << FastPrimeFactorization::is_prime(x) << endl; } }