use std::io::Read; fn get_word() -> String { let stdin = std::io::stdin(); let mut stdin=stdin.lock(); let mut u8b: [u8; 1] = [0]; loop { let mut buf: Vec = Vec::with_capacity(16); loop { let res = stdin.read(&mut u8b); if res.unwrap_or(0) == 0 || u8b[0] <= b' ' { break; } else { buf.push(u8b[0]); } } if buf.len() >= 1 { let ret = String::from_utf8(buf).unwrap(); return ret; } } } fn get() -> T { get_word().parse().ok().unwrap() } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl Default for ModInt { fn default() -> Self { Self::new_internal(0) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl ::std::fmt::Debug for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { let (mut a, mut b, _) = red(self.x, M::m()); if b < 0 { a = -a; b = -b; } write!(f, "{}/{}", a, b) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } // Finds the simplest fraction x/y congruent to r mod p. // The return value (x, y, z) satisfies x = y * r + z * p. fn red(r: i64, p: i64) -> (i64, i64, i64) { if r.abs() <= 10000 { return (r, 1, 0); } let mut nxt_r = p % r; let mut q = p / r; if 2 * nxt_r >= r { nxt_r -= r; q += 1; } if 2 * nxt_r <= -r { nxt_r += r; q -= 1; } let (x, z, y) = red(nxt_r, r); (x, y - q * z, z) } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] pub struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 998_244_353; define_mod!(P, MOD); type MInt = mod_int::ModInt

; // https://judge.yosupo.jp/submission/5155 mod pollard_rho { /// binary gcd pub fn gcd(mut x: i64, mut y: i64) -> i64 { if y == 0 { return x; } if x == 0 { return y; } let k = (x | y).trailing_zeros(); y >>= k; x >>= x.trailing_zeros(); while y != 0 { y >>= y.trailing_zeros(); if x > y { let t = x; x = y; y = t; } y -= x; } x << k } fn add_mod(x: i64, y: i64, n: i64) -> i64 { let z = x + y; if z >= n { z - n } else { z } } fn mul_mod(x: i64, mut y: i64, n: i64) -> i64 { assert!(x >= 0); assert!(x < n); let mut sum = 0; let mut cur = x; while y > 0 { if (y & 1) == 1 { sum = add_mod(sum, cur, n); } cur = add_mod(cur, cur, n); y >>= 1; } sum } fn mod_pow(x: i64, mut e: i64, n: i64) -> i64 { let mut prod = if n == 1 { 0 } else { 1 }; let mut cur = x % n; while e > 0 { if (e & 1) == 1 { prod = mul_mod(prod, cur, n); } e >>= 1; if e > 0 { cur = mul_mod(cur, cur, n); } } prod } pub fn is_prime(n: i64) -> bool { if n <= 1 { return false; } let small = [2, 3, 5, 7, 11, 13]; if small.iter().any(|&u| u == n) { return true; } if small.iter().any(|&u| n % u == 0) { return false; } let mut d = n - 1; let e = d.trailing_zeros(); d >>= e; // https://miller-rabin.appspot.com/ let a = [2, 325, 9375, 28178, 450775, 9780504, 1795265022]; a.iter().all(|&a| { if a % n == 0 { return true; } let mut x = mod_pow(a, d, n); if x == 1 { return true; } for _ in 0..e { if x == n - 1 { return true; } x = mul_mod(x, x, n); if x == 1 { return false; } } x == 1 }) } fn pollard_rho(n: i64, c: &mut i64) -> i64 { // An improvement with Brent's cycle detection algorithm is performed. // https://maths-people.anu.edu.au/~brent/pub/pub051.html if n % 2 == 0 { return 2; } loop { let mut x: i64; // tortoise let mut y = 2; // hare let mut d = 1; let cc = *c; let f = |i| add_mod(mul_mod(i, i, n), cc, n); let mut r = 1; // We don't perform the gcd-once-in-a-while optimization // because the plain gcd-every-time algorithm appears to // outperform, at least on judge.yosupo.jp :) while d == 1 { x = y; for _ in 0..r { y = f(y); d = gcd((x - y).abs(), n); if d != 1 { break; } } r *= 2; } if d == n { *c += 1; continue; } return d; } } /// Outputs (p, e) in p's ascending order. pub fn factorize(x: i64) -> Vec<(i64, usize)> { if x <= 1 { return vec![]; } let mut hm = std::collections::HashMap::new(); let mut pool = vec![x]; let mut c = 1; while let Some(u) = pool.pop() { if is_prime(u) { *hm.entry(u).or_insert(0) += 1; continue; } let p = pollard_rho(u, &mut c); pool.push(p); pool.push(u / p); } let mut v: Vec<_> = hm.into_iter().collect(); v.sort(); v } } // mod pollard_rho // https://yukicoder.me/problems/no/2578 (4) // Solved with hints // m <= 10^18 なので m の約数は 10^5 個程度あるので、約数をキーに持つ単純な DP だとうまくいかない。 // g(t) := \sum{lcm | t} \prod W とすると、g(t) = \prod{a_i | t} (W_i + 1) である。(足すものが乗法的であるため分配法則が使える。) // {m の約数} のように線型束の直積になっている束について、包除原理は (1 -1 0 0 ...) の直積である。 // つまり、 f(m) = \sum_{u|m, u is squarefree} (-1)^{#primes(u)} g(t/u) である。 // これによって、2^(m の素因数の個数) で手間を抑えられるので、高速に計算できる。 // https://gist.github.com/koba-e964/2de3a6480749241f424c4e110a440503 によれば m の素因数の個数は 15 個以下であるため、2^15 * n 程度の手間である。 // -> TLE。高速ゼータ変換を使って (m の素因数の個数)n にする必要があった。 // Tags: product-of-lattices, inclusion-exclusion-principle fn main() { let t: usize = get(); let m: i64 = get(); let pe = pollard_rho::factorize(m); let l = pe.len(); for _ in 0..t { let n: usize = get(); let b: i64 = get(); let c: i64 = get(); let d: i64 = get(); let a: Vec = (0..n).map(|_| get()).collect(); let mut w = vec![MInt::new(b); n]; for i in 1..n { w[i] = w[i - 1] * c + d; } let mut dp = vec![MInt::new(1); 1 << l]; for i in 0..n { if m % a[i] != 0 { continue; } let rem = m / a[i]; let mut bits = 0; for j in 0..l { if rem % pe[j].0 == 0 { bits |= 1 << j; } } dp[bits] *= w[i] + 1; } for i in 0..l { for bits in 0usize..1 << l { if (bits & 1 << i) == 0 { dp[bits] = dp[bits] * dp[bits ^ 1 << i]; } } } let mut tot = MInt::new(0); for bits in 0..1usize << l { if bits.count_ones() % 2 == 0 { tot += dp[bits]; } else { tot -= dp[bits]; } } if m == 1 { tot -= 1; // exclude the empty set } println!("{}", tot); } }