結果

問題 No.2487 Multiple of M
ユーザー koba-e964koba-e964
提出日時 2023-10-01 02:37:13
言語 Rust
(1.83.0 + proconio)
結果
AC  
実行時間 1 ms / 2,000 ms
コード長 6,534 bytes
コンパイル時間 12,431 ms
コンパイル使用メモリ 390,156 KB
実行使用メモリ 6,948 KB
最終ジャッジ日時 2024-07-23 20:05:44
合計ジャッジ時間 14,593 ms
ジャッジサーバーID
(参考情報)
judge5 / judge4
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 3
other AC * 53
権限があれば一括ダウンロードができます

ソースコード

diff #
プレゼンテーションモードにする

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<u8> = 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: std::str::FromStr>() -> 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<M> { pub x: i64, phantom: ::std::marker::PhantomData<M> }
impl<M: Mod> ModInt<M> {
// 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<M: Mod> Default for ModInt<M> {
fn default() -> Self { Self::new_internal(0) }
}
impl<M: Mod, T: Into<ModInt<M>>> Add<T> for ModInt<M> {
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<M: Mod, T: Into<ModInt<M>>> Sub<T> for ModInt<M> {
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<M: Mod, T: Into<ModInt<M>>> Mul<T> for ModInt<M> {
type Output = Self;
fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) }
}
impl<M: Mod, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
fn add_assign(&mut self, other: T) { *self = *self + other; }
}
impl<M: Mod, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
fn sub_assign(&mut self, other: T) { *self = *self - other; }
}
impl<M: Mod, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
fn mul_assign(&mut self, other: T) { *self = *self * other; }
}
impl<M: Mod> Neg for ModInt<M> {
type Output = Self;
fn neg(self) -> Self { ModInt::new(0) - self }
}
impl<M> ::std::fmt::Display for ModInt<M> {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
self.x.fmt(f)
}
}
impl<M: Mod> ::std::fmt::Debug for ModInt<M> {
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<M: Mod> From<i64> for ModInt<M> {
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<P>;
fn gcd(mut a: i64, mut b: i64) -> i64 {
while b != 0 {
let r = a % b;
a = b;
b = r;
}
a
}
// https://yukicoder.me/problems/no/2487 (3.5)
// 使f(S) := (0 <= A_i <= M-1, x in S A_x = 0 )
// m = min U\S m f(S)(-1)^{|S|} (-1)^m(M-1)^{N-1-m}M/x_m
// x_i x_0 = M, x_{i+1} = x_i/gcd(x_i, K)
// x_m x_m
// \sum_{0 <= m <= N-1} (-1)^m(M-1)^{N-1-m}M (M-2)^{N-1}M
// x_m (-1)^m(M-1)^{N-1-m}M/x_m
// S = U min U\S f(U) = 1
// -> (-1)^m(M-1)^{N-1-m} (M-2)^{N-1} ()
// (M-1)^{N-1}(1 - (-1/(M-1))^N)/(1 + 1/(M-1)) = ((M-1)^N - (-1)^N) / M
// -> |x| > n n AC
fn main() {
let n: i64 = get();
let m: i64 = get();
let k: i64 = get();
let mut x = vec![m];
let mut c = m;
loop {
let oldc = c;
c = c / gcd(c, k);
if oldc == c { break; }
x.push(c);
}
let mut tot = (MInt::new(m - 1).pow(n) - MInt::new(MOD - 1).pow(n)) * MInt::new(c).inv();
for i in 0..std::cmp::min(x.len() as i64, n) {
let mut num = MInt::new(m - 1).pow(n - 1 - i) * m;
if i % 2 != 0 {
num = -num;
}
tot += num * MInt::new(x[i as usize]).inv();
tot -= num * MInt::new(c).inv();
}
if n % 2 != 0 {
tot -= 1;
} else {
tot += 1;
}
println!("{}", tot);
}
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