use proconio::{fastout, input}; const MODULUS: i128 = 10_i128.pow(9); #[fastout] fn main() { input! { t: usize, nm: [(i128, i128); t], } for &(n, m) in &nm { match solve(n, m) { Some(ans) => println!("{}", ans), None => println!("-1"), } } } fn solve(n: i128, m: i128) -> Option<i128> { let a = n; let b = MODULUS; let c = -m; let (x0, _, d) = ext_gcd(a, b); if c % d != 0 { return None; } let scale = c / d; let scaled_x = x0 * scale; let divided_b = b / d; let mut best_x = (scaled_x % divided_b + divided_b) % divided_b; if best_x == 0 { best_x = divided_b; } Some(best_x) } pub fn modinv(mut a: i128, m: i128) -> i128 { assert!(m >= 2); let (mut b, mut s, mut t) = (m, 1, 0); while b != 0 { let q = a / b; a -= q * b; std::mem::swap(&mut a, &mut b); s -= q * t; std::mem::swap(&mut s, &mut t); } assert_eq!( a.abs(), 1, "The inverse does not exist. gcd(a, m) = {}", a.abs() ); s %= m; if s < 0 { s += m; } s } /// Returns a tuple of `(x, y)` and `gcd(a, b)` that satisfy `ax + by = gcd(a, b)` in that order. /// /// The returned `x`, `y` and `gcd(a, b)` satisfy the following: /// * Both `|x|` and `|y|` are less than or equal to `max(|a|, |b|)`. /// * `gcd(a, b)` is non-negative. pub fn ext_gcd(mut a: i128, mut b: i128) -> (i128, i128, i128) { if a == 0 && b == 0 { return (0, 0, 0); } let (mut s, mut t, mut u, mut v) = (1, 0, 0, 1); while b != 0 { (a, b, s, t, u, v) = (b, a % b, t, s - a / b * t, v, u - a / b * v); } let sgn = a.signum(); (sgn * s, sgn * u, sgn * a) }