/** * code generated by JHelper * More info: https://github.com/AlexeyDmitriev/JHelper * @author */ #include using namespace std; using ll = long long; using ld = long double; template using P = pair; template using V = vector; using VI = vector; using VL = vector; //#pragma GCC optimize("O3") //#pragma GCC target("avx2") //#pragma GCC target("avx512f") //#pragma GCC optimize("unroll-loops") //#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native") //#pragma GCC optimize("Ofast") #define G(size_1) vector>(size_1, vector()) #define SZ(x) ((long long)(x).size()) #define READ ({long long t;cin >> t;t;}) #define FOR(i, __begin, __end) for (auto i = (__begin) - ((__begin) > (__end)); i != (__end) - ((__begin) > (__end)); i += 1 - 2 * ((__begin) > (__end))) #define REP(i, __end) for (auto i = decltype(__end){0}; i < (__end); ++i) #define ALL(x) (x).begin(),(x).end() #define RALL(x) (x).rbegin(),(x).rend() #define F first #define S second #define y0 y3487465 #define y1 y8687969 #define j0 j1347829 #define j1 j234892 #define BIT(n) (1LL<<(n)) #define UNIQUE(v) v.erase( unique(v.begin(), v.end()), v.end() ) #define EB emplace_back #define PB push_back #define fcout cout << fixed << setprecision(12) #define fcerr cerr << fixed << setprecision(12) #define print(x) cout << (x) << '\n' #define printE(x) cout << (x) << endl; #define fprint(x) cout << fixed << setprecision(12) << (x) << '\n' # define BYE(a) do { cout << (a) << endl; return ; } while (false) #define LB lower_bound #define UB upper_bound #define LBI(c, x) distance((c).begin(), lower_bound((c).begin(), (c).end(), (x))) #define UBI(c, x) distance((c).begin(), upper_bound((c).begin(), (c).end(), (x))) #ifdef DEBUG #define DBG(args...) { string _s = #args; replace(_s.begin(), _s.end(), ',', ' '); stringstream _ss(_s); istream_iterator _it(_ss); _err(cerr,_it, args); } #define ERR(args...) { string _s = #args; replace(_s.begin(), _s.end(), ',', ' '); stringstream _ss(_s); istream_iterator _it(_ss); _err(std::cerr,_it, args); } #else #define DBG(args...) {}; #define ERR(args...) {}; #endif void _err(std::ostream& cerr, istream_iterator it){cerr << endl;} template void _err(std::ostream& cerr, istream_iterator it, T a, Args... args){ cerr << *it << " = " << a << " "; _err(cerr, ++it, args...); } namespace aux{ template struct seq{ }; template struct gen_seq : gen_seq{ }; template struct gen_seq<0, Is...> : seq{ }; template void print_tuple(std::basic_ostream& os, Tuple const& t, seq){ using swallow = int[]; (void) swallow{0, (void(os << (Is == 0 ? "" : ",") << std::get(t)), 0)...}; } template void read_tuple(std::basic_istream& os, Tuple& t, seq){ using swallow = int[]; (void) swallow{0, (void(os >> std::get(t)), 0)...}; } } // aux:: template auto operator<<(std::basic_ostream& os, std::tuple const& t) -> std::basic_ostream&{ os << "("; aux::print_tuple(os, t, aux::gen_seq()); return os << ")"; } template auto operator>>(std::basic_istream& os, std::tuple& t) -> std::basic_istream&{ aux::read_tuple(os, t, aux::gen_seq()); return os; } template inline bool chmax(T& a, const T& b){ if(a < b){ a = b; return 1; } return 0; } template inline bool chmin(T& a, const T& b){ if(b < a){ a = b; return 1; } return 0; } template istream& operator>>(istream& is, pair& V){ is >> V.F >> V.S; return is; } template istream& operator>>(istream& is, vector& V){ for(auto&& ele : V)is >> ele; return is; } template ostream& operator<<(ostream& os, const vector V){ os << "["; int cnt = 0; T curr; if(!V.empty()){ for(int i = 0; i < V.size() - 1; ++i){ if(V[i] == curr)cnt++; else cnt = 0; if(cnt == 4)os << "... "; if(cnt < 4) os << i << ":" << V[i] << " "; curr = V[i]; } os << V.size() - 1 << ":" << V.back(); } os << "]\n"; return os; } template ostream& operator<<(ostream& os, const pair P){ os << "("; os << P.first << "," << P.second; os << ")"; return os; } template ostream& operator<<(ostream& os, const set V){ os << "{"; if(!V.empty()){ auto it = V.begin(); for(int i = 0; i < V.size() - 1; ++i){ os << *it << " "; it++; } os << *it; } os << "}\n"; return os; } template ostream& operator<<(ostream& os, const unordered_set V){ os << "{"; if(!V.empty()){ auto it = V.begin(); for(int i = 0; i < V.size() - 1; ++i){ os << *it << " "; it++; } os << *it; } os << "}\n"; return os; } template ostream& operator<<(ostream& os, const multiset V){ os << "{"; if(!V.empty()){ auto it = V.begin(); for(int i = 0; i < V.size() - 1; ++i){ os << *it << " "; it++; } os << *it; } os << "}"; return os; } template ostream& operator<<(ostream& os, const map V){ os << "{"; if(!V.empty()){ auto it = V.begin(); for(int i = 0; i < V.size() - 1; ++i){ os << "("; os << it->first << "," << it->second; os << ") "; it++; } os << "("; os << it->first << "," << it->second; os << ")"; } os << "}\n"; return os; } template ostream& operator<<(ostream& os, const unordered_map V){ os << "{"; if(!V.empty()){ auto it = V.begin(); for(int i = 0; i < V.size() - 1; ++i){ os << "("; os << it->first << "," << it->second; os << ") "; it++; } os << "("; os << it->first << "," << it->second; os << ")"; } os << "}\n"; return os; } template ostream& operator<<(ostream& os, const deque V){ os << "["; if(!V.empty()){ for(int i = 0; i < V.size() - 1; ++i){ os << V[i] << "->"; } if(!V.empty())os << V.back(); } os << "]\n"; return os; }; template ostream& operator<<(ostream& os, const priority_queue V){ priority_queue _V = V; os << "["; if(!_V.empty()){ while(_V.size() > 1){ os << _V.top() << "->"; _V.pop(); } os << _V.top(); } os << "]\n"; return os; }; template struct y_combinator{ F f; // the lambda will be stored here // a forwarding operator(): template decltype(auto) operator()(Args&& ... args) const{ // we pass ourselves to f, then the arguments. // the lambda should take the first argument as `auto&& recurse` or similar. return f(*this, std::forward(args)...); } }; // helper function that deduces the type of the lambda: template y_combinator> recursive(F&& f){ return {std::forward(f)}; } struct hash_pair{ template size_t operator()(const pair& p) const{ auto hash1 = hash{}(p.first); auto hash2 = hash{}(p.second); return hash1^hash2; } }; template auto vec(int n, U v){ return std::vector(n, v); } template auto vec(int n, Args... args){ auto val = vec(std::forward(args)...); return std::vector(n, std::move(val)); } const double PI = 2*acos(.0); const int INF = 0x3f3f3f3f; template inline T ceil(T a, T b){return (a + b - 1)/b;} inline long long popcount(ll x){return __builtin_popcountll(x);} int64_t mod_log(int64_t a, int64_t b, int64_t p){ int64_t g = 1; for(int64_t i = p; i; i /= 2) (g *= a) %= p; g = __gcd(g, p); int64_t t = 1, c = 0; for(; t%g; c++){ if(t == b) return c; (t *= a) %= p; } if(b%g) return -1; t /= g; b /= g; int64_t n = p/g, h = 0, gs = 1; for(; h*h < n; h++) (gs *= a) %= n; unordered_map bs; for(int64_t s = 0, e = b; s < h; bs[e] = ++s){ (e *= a) %= n; } for(int64_t s = 0, e = t; s < n;){ (e *= gs) %= n; s += h; if(bs.count(e)) return c + s - bs[e]; } return -1; } #ifndef ATCODER_MATH_HPP #define ATCODER_MATH_HPP 1 #include #include #include #include #ifndef ATCODER_INTERNAL_MATH_HPP #define ATCODER_INTERNAL_MATH_HPP 1 #include namespace atcoder { namespace internal { // @param m `1 <= m` // @return x mod m constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } // Fast moduler by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; unsigned long long im; // @param m `1 <= m` barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { // [1] m = 1 // a = b = im = 0, so okay // [2] m >= 2 // im = ceil(2^64 / m) // -> im * m = 2^64 + r (0 <= r < m) // let z = a*b = c*m + d (0 <= c, d < m) // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2 // ((ab * im) >> 64) == c or c + 1 unsigned long long z = a; z *= b; #ifdef _MSC_VER unsigned long long x; _umul128(z, im, &x); #else unsigned long long x = (unsigned long long)(((unsigned __int128)(z)*im) >> 64); #endif unsigned int v = (unsigned int)(z - x * _m); if (_m <= v) v += _m; return v; } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; long long d = n - 1; while (d % 2 == 0) d /= 2; for (long long a : {2, 7, 61}) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template constexpr bool is_prime = is_prime_constexpr(n); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; // Contracts: // [1] s - m0 * a = 0 (mod b) // [2] t - m1 * a = 0 (mod b) // [3] s * |m1| + t * |m0| <= b long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b // [3]: // (s - t * u) * |m1| + t * |m0 - m1 * u| // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) // = s * |m1| + t * |m0| <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } // by [3]: |m0| <= b/g // by g != b: |m0| < b/g if (m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (long long)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template constexpr int primitive_root = primitive_root_constexpr(m); } // namespace internal } // namespace atcoder #endif // ATCODER_INTERNAL_MATH_HPP namespace atcoder { long long pow_mod(long long x, long long n, int m) { assert(0 <= n && 1 <= m); if (m == 1) return 0; internal::barrett bt((unsigned int)(m)); unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m)); while (n) { if (n & 1) r = bt.mul(r, y); y = bt.mul(y, y); n >>= 1; } return r; } long long inv_mod(long long x, long long m) { assert(1 <= m); auto z = internal::inv_gcd(x, m); assert(z.first == 1); return z.second; } // (rem, mod) std::pair crt(const std::vector& r, const std::vector& m) { assert(r.size() == m.size()); int n = int(r.size()); // Contracts: 0 <= r0 < m0 long long r0 = 0, m0 = 1; for (int i = 0; i < n; i++) { assert(1 <= m[i]); long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i]; if (m0 < m1) { std::swap(r0, r1); std::swap(m0, m1); } if (m0 % m1 == 0) { if (r0 % m1 != r1) return {0, 0}; continue; } // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1) // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1)); // r2 % m0 = r0 // r2 % m1 = r1 // -> (r0 + x*m0) % m1 = r1 // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1) // -> x = (r1 - r0) / g * inv(u0) (mod u1) // im = inv(u0) (mod u1) (0 <= im < u1) long long g, im; std::tie(g, im) = internal::inv_gcd(m0, m1); long long u1 = (m1 / g); // |r1 - r0| < (m0 + m1) <= lcm(m0, m1) if ((r1 - r0) % g) return {0, 0}; // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1) long long x = (r1 - r0) / g % u1 * im % u1; // |r0| + |m0 * x| // < m0 + m0 * (u1 - 1) // = m0 + m0 * m1 / g - m0 // = lcm(m0, m1) r0 += x * m0; m0 *= u1; // -> lcm(m0, m1) if (r0 < 0) r0 += m0; } return {r0, m0}; } long long floor_sum(long long n, long long m, long long a, long long b) { long long ans = 0; if (a >= m) { ans += (n - 1) * n * (a / m) / 2; a %= m; } if (b >= m) { ans += n * (b / m); b %= m; } long long y_max = (a * n + b) / m, x_max = (y_max * m - b); if (y_max == 0) return ans; ans += (n - (x_max + a - 1) / a) * y_max; ans += floor_sum(y_max, a, m, (a - x_max % a) % a); return ans; } } // namespace atcoder #endif // ATCODER_MATH_HPP using namespace atcoder; void solve(std::istream& cin, std::ostream& cout, std::ostream& cerr){ auto solve = [&](){ ll N; cin >> N; while(N%2==0)N/=2; while(N%5==0)N/=5; // ll g = gcd(10, N); // ll t = atcoder::inv_mod(10/gcd(10,N), N/gcd(10,N)); if(ll x = mod_log(10, inv_mod(10, N), N);x)print(x + 1); else print(1); }; int testcases; cin >> testcases; for(int case_num = 1; case_num <= testcases; case_num++){ solve(); } } #undef int int main() { istream& in(cin); ostream& out(cout); ostringstream err; in.tie(0); ios::sync_with_stdio(0); solve(in, out, err); return 0; }