#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; #define int long long #define ii pair #define app push_back #define all(a) a.begin(), a.end() #define bp __builtin_popcountll #define ll long long #define mp make_pair #define x first #define y second #define Time (double)clock()/CLOCKS_PER_SEC #define debug(x) std::cerr << #x << ": " << x << '\n'; #define FOR(i, n) for (int i = 0; i < n; ++i) #define pb push_back #define trav(a, x) for (auto& a : x) using vi = vector; template std::ostream& operator <<(std::ostream& output, const pair & data) { output << "(" << data.x << "," << data.y << ")"; return output; } template std::ostream& operator <<(std::ostream& output, const std::vector& data) { for (const T& x : data) output << x << " "; return output; } ll div_up(ll a, ll b) { return a/b+((a^b)>0&&a%b); } // divide a by b rounded up ll div_down(ll a, ll b) { return a/b-((a^b)<0&&a%b); } // divide a by b rounded down ll math_mod(ll a, ll b) { return a - b * div_down(a, b); } #define tcT template using V = vector; tcT> void re(V& x) { trav(a, x) cin >> a; } tcT> bool ckmin(T& a, const T& b) { return b < a ? a = b, 1 : 0; } // set a = min(a,b) tcT> bool ckmax(T& a, const T& b) { return a < b ? a = b, 1 : 0; } ll gcd(ll a, ll b) { while (b) { tie(a, b) = mp(b, a % b); } return a; } int powmod(int a, int b, int p) { int res = 1; while (b > 0) { if (b & 1) { res = res * a % p; } a = a * a % p; b >>= 1; } return res; } // Finds the primitive root modulo p int generator(int p) { vector fact; int phi = p-1, n = phi; for (int i = 2; i * i <= n; ++i) { if (n % i == 0) { fact.push_back(i); while (n % i == 0) n /= i; } } if (n > 1) fact.push_back(n); for (int res = 2; res <= p; ++res) { bool ok = true; for (int factor : fact) { if (powmod(res, phi / factor, p) == 1) { ok = false; break; } } if (ok) return res; } return -1; } // This program finds all numbers x such that x^k = a (mod n) void solve(int n, int k) { int a = 1; int g = generator(n); // Baby-step giant-step discrete logarithm algorithm int sq = (int) sqrt (n + .0) + 1; vector> dec(sq); for (int i = 1; i <= sq; ++i) dec[i-1] = {powmod(g, i * sq * k % (n - 1), n), i}; sort(dec.begin(), dec.end()); int any_ans = -1; for (int i = 0; i < sq; ++i) { int my = powmod(g, i * k % (n - 1), n) * a % n; auto it = lower_bound(dec.begin(), dec.end(), make_pair(my, 0ll)); if (it != dec.end() && it->first == my) { any_ans = it->second * sq - i; break; } } if (any_ans == -1) { assert(0); } // Print all possible answers int delta = (n-1) / gcd(k, n-1); vector ans; for (int cur = any_ans % delta; cur < n-1; cur += delta) ans.push_back(powmod(g, cur, n)); sort(ans.begin(), ans.end()); cout << ans << endl; } signed main() { #ifdef ONLINE_JUDGE #define endl '\n' ios_base::sync_with_stdio(0); cin.tie(0); #endif int t; cin >> t; while (t--) { int v, x; cin >> v >> x; solve(v * x + 1, x); } }