#![allow(unused_imports,non_snake_case,dead_code)] use std::{cmp::Reverse as Rev,collections::*,iter::*}; use proconio::{marker::*,*}; // use rand::prelude::*; #[fastout] fn main(){ input!{ qs:[(usize,usize)], } // let mut rng=StdRng::seed_from_u64(0); // let mut qs=(0..1000).map(|_|(rng.gen_range(1..=100),rng.gen_range(1..=100))).collect::>(); let M=|a|M::new(a); let mut path=vec![]; for &(n,a) in &qs{ if a==1{ println!("{}",M(n)*(n-1)/2); continue; } path.clear(); path.push(n); let mut ans=M(0); let mut last=n; while last!=0{ if last%a!=0{ last=last/a*a; } else{ last/=a; } path.push(last); } for w in path.windows(2){ if w[0]%a!=0{ ans+=M(w[0]-1+w[1])*(w[0]-w[1])/2; } else{ let len=w[0]-w[1]-1; ans+=M(len)*(len+1)/2; ans+=w[1]; } } // let verify=solve(n,a); // assert!(verify==ans); // if verify!=ans{ // eprintln!("{} {}",n,a); // eprintln!("verify = {verify}"); // eprintln!("ans = {ans}"); // } println!("{ans}"); } } fn solve(n:usize,a:usize)->M{ let mut dp=vec![usize::MAX/64;n+1]; dp[n]=0; let mut sum=M::new(0); for i in (1..=n).rev(){ sum+=M::new(dp[i]); dp[i-1]=dp[i-1].min(dp[i]+1); if i%a==0{ dp[i/a]=dp[i/a].min(dp[i]+1); } } sum } type M = StaticModInt<998244353>; type ModInt1000000007 = StaticModInt<1000000007>; #[derive(Clone, Copy, PartialEq, Eq, Default, Hash)] struct StaticModInt { val: u32, } impl ModIntBase for StaticModInt { fn modulus() -> u32 { MODULO } unsafe fn raw(val: u32) -> Self { StaticModInt { val } } fn val(self) -> u32 { self.val } fn inv(self) -> Self { assert!(self.val != 0, "divisor by zero"); let inv_val = mod_inv_by_ext_gcd(self.val, MODULO); unsafe { Self::raw(inv_val) } } } impl std::fmt::Display for StaticModInt { fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { write!(f, "{}", self.val) } } impl std::fmt::Debug for StaticModInt { fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { write!(f, "{}", self.val) } } macro_rules! impl_from_int { ($($t:ty),*) => { $(impl From<$t> for StaticModInt { fn from(x: $t) -> Self { StaticModInt::new(x) } })* } } impl_from_int!(i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize); impl std::ops::Add for StaticModInt { type Output = Self; fn add(self, rhs: Self) -> Self::Output { let mut sum = self.val + rhs.val; if sum >= MODULO { sum -= MODULO; } unsafe { Self::raw(sum) } } } impl std::ops::Sub for StaticModInt { type Output = Self; fn sub(self, rhs: Self) -> Self::Output { let diff = if self.val >= rhs.val { self.val - rhs.val } else { MODULO + self.val - rhs.val }; unsafe { Self::raw(diff) } } } impl std::ops::Mul for StaticModInt { type Output = Self; fn mul(self, rhs: Self) -> Self::Output { let prod = (self.val as u64 * rhs.val as u64 % MODULO as u64) as u32; unsafe { Self::raw(prod) } } } impl std::ops::Div for StaticModInt { type Output = Self; fn div(self, rhs: Self) -> Self::Output { self * rhs.inv() } } impl std::ops::AddAssign for StaticModInt { fn add_assign(&mut self, rhs: Self) { *self = *self + rhs; } } impl std::ops::SubAssign for StaticModInt { fn sub_assign(&mut self, rhs: Self) { *self = *self - rhs; } } impl std::ops::MulAssign for StaticModInt { fn mul_assign(&mut self, rhs: Self) { *self = *self * rhs; } } impl std::ops::DivAssign for StaticModInt { fn div_assign(&mut self, rhs: Self) { *self = *self / rhs; } } impl std::ops::Neg for StaticModInt { type Output = Self; fn neg(self) -> Self::Output { if self.val == 0 { self } else { unsafe { Self::raw(MODULO - self.val) } } } } impl std::str::FromStr for StaticModInt { type Err = ::Err; fn from_str(s: &str) -> Result { let x = s.parse::()?; Ok(StaticModInt::new(x)) } } macro_rules! impl_mod_int_ops_sub { ($trait:ident, $func:ident, $assign_trait:ident, $assign_func:ident, $op:tt, ($($t:ty),*)) => { $( impl std::ops::$trait<$t> for StaticModInt { type Output = Self; fn $func(self, rhs: $t) -> Self::Output { self $op StaticModInt::new(rhs) } } impl std::ops::$trait> for $t { type Output = StaticModInt; fn $func(self, rhs: StaticModInt) -> Self::Output { StaticModInt::new(self) $op rhs } } impl std::ops::$assign_trait<$t> for StaticModInt { fn $assign_func(&mut self, rhs: $t) { *self = *self $op StaticModInt::new(rhs); } } )* } } macro_rules! impl_mod_int_ops { ($($t:tt)*) => { impl_mod_int_ops_sub!($($t)*, (i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize)); } } impl_mod_int_ops!(Add, add, AddAssign, add_assign, +); impl_mod_int_ops!(Sub, sub, SubAssign, sub_assign, -); impl_mod_int_ops!(Mul, mul, MulAssign, mul_assign, *); impl_mod_int_ops!(Div, div, DivAssign, div_assign, /); impl std::iter::Sum for StaticModInt { fn sum>(iter: I) -> Self { iter.fold(StaticModInt::zero(), |acc, x| acc + x) } } impl std::iter::Product for StaticModInt { fn product>(iter: I) -> Self { iter.fold(StaticModInt::one(), |acc, x| acc * x) } } trait RemEuclidU32 { fn rem_euclid_u32(self, modulus: u32) -> u32; } macro_rules! impl_rem_euclid_u32_for_small_signed { ($($ty:tt),*) => { $( impl RemEuclidU32 for $ty { fn rem_euclid_u32(self, modulus: u32) -> u32 { (self as i64).rem_euclid(i64::from(modulus)) as _ } } )* } } impl_rem_euclid_u32_for_small_signed!(i8, i16, i32, i64, isize); impl RemEuclidU32 for i128 { fn rem_euclid_u32(self, modulus: u32) -> u32 { self.rem_euclid(i128::from(modulus)) as _ } } macro_rules! impl_rem_euclid_u32_for_small_unsigned { ($($ty:tt),*) => { $( impl RemEuclidU32 for $ty { fn rem_euclid_u32(self, modulus: u32) -> u32 { self as u32 % modulus } } )* } } macro_rules! impl_rem_euclid_u32_for_large_unsigned { ($($ty:tt),*) => { $( impl RemEuclidU32 for $ty { fn rem_euclid_u32(self, modulus: u32) -> u32 { (self % (modulus as $ty)) as _ } } )* } } impl_rem_euclid_u32_for_small_unsigned!(u8, u16, u32); impl_rem_euclid_u32_for_large_unsigned!(u64, u128); #[cfg(target_pointer_width = "32")] impl_rem_euclid_u32_for_small_unsigned!(usize); #[cfg(target_pointer_width = "64")] impl_rem_euclid_u32_for_large_unsigned!(usize); trait ModIntBase: Default + std::str::FromStr + From + From + From + From + From + From + From + From + From + From + From + From + Copy + Eq + std::hash::Hash + std::fmt::Display + std::fmt::Debug + std::ops::Neg + std::ops::Add + std::ops::Sub + std::ops::Mul + std::ops::Div + std::ops::AddAssign + std::ops::SubAssign + std::ops::MulAssign + std::ops::DivAssign { fn modulus() -> u32; unsafe fn raw(val: u32) -> Self; fn val(self) -> u32; fn inv(self) -> Self; fn new(val: impl RemEuclidU32) -> Self { unsafe { Self::raw(val.rem_euclid_u32(Self::modulus())) } } fn zero() -> Self { unsafe { Self::raw(0) } } fn one() -> Self { unsafe { Self::raw(1) } } fn pow(self, mut n: u64) -> Self { let mut x = self; let mut r = Self::one(); while n > 0 { if n & 1 == 1 { r *= x; } x *= x; n >>= 1; } r } } fn mod_inv_by_ext_gcd(x: u32, modulus: u32) -> u32 { let (mut a, mut b) = (x as i64, modulus as i64); let (mut u, mut v) = (1i64, 0i64); while b != 0 { let t = a / b; a -= t * b; (a,b)=(b,a); u -= t * v; (u,v)=(v,u); } assert!(a==1); ((u % modulus as i64 + modulus as i64) % modulus as i64) as u32 }