#include #include #include #include #include using namespace std; const int LIM = (int)1e9 + 1; const bool READ_PRIMES_FROM_FILE = true; vector primes; void load_primes() { primes.reserve(51000000); if (READ_PRIMES_FROM_FILE) { ifstream fin("primes.txt"); for (int p; fin >> p;) { primes.push_back(p); } } else { bitset sieve; int lim = sqrt(LIM); for (int i = 3; i < lim; i += 2) { if (!sieve[i] && i % 2 == 1) { for (int j = i * 2; j < LIM; j += i) { sieve.set(j); } } } primes.push_back(2); for (int i = 3; i <= LIM; i += 2) { if (!sieve[i]) { primes.push_back(i); } } } } // Returns k in the p^k term of the prime factorization of N!. int legendre_exp(int p, int N) { int ans = 0; for (long long base = p; base <= N; base *= p) { ans += N / base; } return ans; } int powmod8(int p, int exp) { int ans = 1; for (exp %= 8; exp > 0; exp--) { ans = (ans * p) % 8; } return ans; } int N, K; int solve() { bool is_one = true, is_two = true, two_exp_even = false; int mod8prod = 1; for (int p : primes) { if (p > N) { break; } int exp = legendre_exp(p, N) - legendre_exp(p, K); if (exp % 2 != 0) { is_one = false; } if (p % 4 == 3 && exp % 2 == 1) { is_two = false; } if (p == 2 && exp % 2 == 0) { two_exp_even = true; } if (p > 2) { mod8prod = (mod8prod * powmod8(p, exp)) % 8; } } bool is_three = !(two_exp_even && mod8prod == 7); return is_one ? 1 : is_two ? 2 : is_three ? 3 : 4; } int main() { ios_base::sync_with_stdio(false); cin.tie(nullptr); load_primes(); int T; cin >> T; for (int t = 1; t <= T; t++) { cin >> N >> K; cout << "Case #" << t << ": " << solve() << endl; } return 0; }