#include using namespace std; using Int8 = int8_t; using Int16 = int16_t; using Int32 = int32_t; using Int64 = int64_t; using Int128 = __int128_t; using Word8 = uint8_t; using Word16 = uint16_t; using Word32 = uint32_t; using Word64 = uint64_t; using Word128 = __uint128_t; using Int = int_fast64_t; using Word = uint_fast64_t; using F32 = float; using F64 = double; using F80 = long double; static inline Word ex_gcd(Word y) { Word u = 1, v = 0; Word x = 1ULL << 63; for (int i = 0; i < 64; i++) { if (u & 1){ u = (u + y) / 2; v = v / 2 + x; } else { u >>= 1; v >>= 1; } } return -v; } int jacobi(Int a, Word n) { Word t; int j = 1; while(a) { if (a < 0) { a = -a ; if ((n & 3) == 3) j = -j; } int ba = __builtin_ctzll(a); a >>= ba; if (((n & 7) == 3 || (n & 7) == 5) && (ba & 1)) j = -j; if ((a & n & 3) == 3) j = -j; t = a; a = n; n = t; a %= n; if (a > n / 2) a -= n; } return n == 1 ? j : 0; } static inline Word addmod64(Word x, Word y, Word n) { return x + y >= n ? x + y - n : x + y; } static inline Word submod64(Word x, Word y, Word n) { return x >= y ? x - y : x - y + n; } static inline Word64 MR(Word128 x, Word64 m, Word64 n) { Int64 z = (x >> 64) - ((((Word64)x * m) * (Word128)n) >> 64); return z < 0 ? z + n : z; } static inline Word64 RM(Word64 x, Word64 r2, Word64 m, Word64 n) { return MR((Word128)r2 * x, m, n); } static inline Word64 mulmod64(Word64 x, Word64 y, Word64 m, Word64 n) { return MR((Word128)x * y, m, n); } int solovay_strassen(const Word64 n) { if(n <= 1) return 0; if(n <= 3) return 1; if(!(n & 1)) return 0; const Word64 one = -1ULL % n + 1; const Word64 r2 = (Word128) (Int128) -1 % n + 1; const Word64 m = ex_gcd(n); { Word64 d = (n - 1) << __builtin_clzll(n-1); Word64 t = one << 1; if (t >= n) t -= n; for (d <<= 1; d; d <<= 1) { t = mulmod64(t, t, m, n); if (d >> 63) { t <<= 1; if (t >= n) t -= n; } } if(t != one){ Word64 x = (n - 1) & -(n - 1); Word64 mone = n - one; for (x >>= 2; t != mone; x >>= 1) { if (x == 0) return 0; t = mulmod64(t, t, m, n); } } } { Int64 D = 5; for(int i = 0; jacobi(D, n) != -1 && i < 64; i++) { if(i == 32){ Word32 k = round(sqrtl(n)); if (k * k == n) return 0; } if (i & 1) D -= 2; else D += 2; D = -D; } Word64 Q = RM(D < 0 ? (1 - D) / 4 % n : n - (D - 1) / 4 % n, r2, m, n); Word64 u, v, Qn; Word64 k = (n + 1) << __builtin_clzll(n + 1); u = one; v = one; Qn = Q; D %= (Int64)n; D = RM(D < 0 ? n + D : D, r2, m, n); for (k <<= 1; k; k <<= 1) { u = mulmod64(u,v,m,n); v = submod64(mulmod64(v,v,m,n), addmod64(Qn,Qn,n), n); Qn = mulmod64(Qn, Qn, m, n); if (k >> 63) { Word64 uu = addmod64(u,v,n); if (uu & 1) uu += n; uu >>= 1; v = addmod64(mulmod64(D,u,m,n), v, n); if (v & 1) v += n; v >>= 1; u = uu; Qn = mulmod64(Qn,Q,m,n); } } if (u == 0 || v == 0) return 1; Word64 x = (n + 1) & ~n; for (x >>= 2; x; x >>= 1) { u = mulmod64(u,v,m,n); v = submod64(mulmod64(v,v,m,n), addmod64(Qn,Qn,n), n); if (v == 0) return 1; Qn = mulmod64(Qn,Qn,m,n); } } return 0; } namespace fastio { static constexpr int SZ = 1 << 17; char ibuf[SZ], obuf[SZ]; int pil = 0, pir = 0, por = 0; struct Pre { char num[40000]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i * 4 + j] = n % 10 + '0'; n /= 10; } } } } constexpr pre; inline void load() { memcpy(ibuf, ibuf + pil, pir - pil); pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin); pil = 0; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } inline void rd(char& c) { c = ibuf[pil++]; } template inline void rd(T& x) { if (pil + 32 > pir) load(); char c; do { c = ibuf[pil++]; } while (c < '-'); bool minus = 0; if (c == '-') { minus = 1; c = ibuf[pil++]; } x = 0; while (c >= '0') { x = x * 10 + (c & 15); c = ibuf[pil++]; } if (minus) x = -x; } inline void wt(char c) { obuf[por++] = c; } template inline void wt(T x) { if (por > SZ - 32) flush(); if (!x) { obuf[por++] = '0'; return; } if (x < 0) { obuf[por++] = '-'; x = -x; } int i = 12; char buf[16]; while (x >= 10000) { memcpy(buf + i, pre.num + (x % 10000) * 4, 4); x /= 10000; i -= 4; } int d = x < 100 ? (x < 10 ? 1 : 2) : (x < 1000 ? 3 : 4); memcpy(obuf + por, pre.num + x * 4 + 4 - d, d); por += d; memcpy(obuf + por, buf + i + 4, 12 - i); por += 12 - i; } struct Dummy { Dummy() { atexit(flush); } } dummy; } // namespace fastio using fastio::rd; using fastio::wt; using namespace std; int main() { cin.tie(nullptr); cout.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); int query; rd(query); while (query--) { Word64 x; rd(x); int ans = solovay_strassen(x); wt(x); wt(' '); wt(ans); wt('\n'); } return 0; }