import typing def extGCD(a: int, b: int) -> typing.Tuple[int, int, int]: '''ax + by = gcd(a, b) を満たす整数解x, yを求める Returns: gcd(a, b), x, y Reference: https://qiita.com/akebono-san/items/f00c0db99342a8d68e5d ''' x, y, u, v = 1, 0, 0, 1 # 単位行列 p, q = a, b while q: k = p // q x -= k * u y -= k * v x, u = u, x y, v = v, y p, q = q, p % q return x * a + y * b, x, y def inv_mod(a: int, m: int) -> int: '''法 m の元での a の逆元を求める Constraints: m と a は互いに素 Returns: x (int): 法 m の元での a の逆元 ※ m と a が互いに素でない場合は-1を返す Notes: ax = 1 (mod m) ax = 1 + -qm (整数q) ax + mq = 1 -> (x, q)の解をextGCDで求める ''' d, x, q = extGCD(a, m) x %= m if x < 0: x -= m return x if d == 1 or d == -1 else -1 def _crt_pair(b1: int, m1: int, b2: int, m2: int) -> typing.Tuple[int, int]: """以下の連立合同式を満たす最小の非負整数 r を求める r = b1 (mod m1) r = b2 (mod m2) Args: b1 (int): r = b1 (mod m1) m1 (int): b2 (int): r = b2 (mod m2) m2 (int): Returns: typing.Tuple[int, int]: R, M (0 <= R < M, M = gcd(m1, m2))を返す。解が存在しないとき(0, -1)を返す。 Notes: > m1x + m2y = d (d = gcd(m1, m2)) を満たすx, yを求める > r = b1 + m1 * x * ((b2 - b1) // d) > (b2 - b1) % d != 0 のとき解は存在しない """ d, x, y = extGCD(m1, m2) if (b2 - b1) % d != 0: R, M = 0, -1 else: c = (b2 - b1) // d M = (m1 * m2) // d R = (m1 * x * c + b1) % M return R, M def crt(b: typing.List[int], m: typing.List[int]) -> typing.Tuple[int, int]: """連立合同式を満たす最小の非負整数 r を求める r = b1 (mod m1) r = b2 (mod m2) ... r = bn (mod mn) Args: b (typing.List[int]): r = bi (mod mi) m (typing.List[int]): r = bi (mod mi) Returns: typing.Tuple[int, int]: R, M (0 <= R < M, M = gcd(m1, m2))を返す。解が存在しないとき(0, -1)を返す。 Reference: https://qiita.com/drken/items/ae02240cd1f8edfc86fd#%E5%95%8F%E9%A1%8C-2yukicoder-0187-%E4%B8%AD%E8%8F%AF%E9%A2%A8-hard Notes: > R = 0 (mod 1) (-> 任意の整数) を初期値として、順に_crt_pairを適用していく """ N = len(b) assert len(m) == N, '!Size of arguments b and m must be equal!' R, M = 0, 1 for i in range(N): R, M = _crt_pair(R, M, b[i], m[i]) if M == -1: break return R, M ''' [ABC186 - E] ax + by = c の一般解 x = b't + c'x' y = -a't + c'y' (a' = a // d, b' = b // d, c' = c // d, x', y': ax + by = gcd(a, b) の解の一つ) ''' def validate_inv_mod(): '''ABC186-E https://atcoder.jp/contests/abc186/tasks/abc186_e ''' from math import gcd T = int(input()) query = [tuple(map(int, input().split())) for _ in range(T)] for N, S, K in query: d = gcd(N, gcd(S, K)) A, B, M = K // d, S // d, N // d x = inv_mod(A, M) if x == -1: print(-1) else: print((-B * x) % M) def validate_crt(): '''ABC193-E https://atcoder.jp/contests/abc193/tasks/abc193_e ''' INF = (1 << 64) T = int(input()) case = [tuple(map(int, input().split())) for _ in range(T)] for X, Y, P, Q in case: Z = 2 * (X + Y) R = P + Q ans = INF for i in range(X, X + Y): for j in range(P, P + Q): r, m = crt([i, j], [Z, R]) if m >= 0: ans = min(ans, r) print(ans if ans < INF else 'infinity') def validate_crt_2(): """yukicoder No.186 https://yukicoder.me/problems/no/186 """ X = [0] * 3 Y = [0] * 3 for i in range(3): X[i], Y[i] = map(int, input().split()) r, m = crt(X, Y) if m == -1: print(-1) else: print(r if r > 0 else m) # validate_inv_mod() # validate_crt() validate_crt_2()