Problem:
Score : 700 points

Problem Statement

Find the smallest possible sum of the digits in the decimal notation of a positive multiple of K.

Constraints

2 \leq K \leq 10^5

K is an integer.

Input

Input is given from Standard Input in the following format:

K

Output

Print the smallest possible sum of the digits in the decimal notation of a positive multiple of K.

Sample Input 1

6

Sample Output 1

3

12=6×2 yields the smallest sum.

Sample Input 2

41

Sample Output 2

5

11111=41×271 yields the smallest sum.

Sample Input 3

79992

Sample Output 3

36

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_macros)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}
 
#[allow(unused_macros)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};
 
    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}
 
#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };
 
    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };
 
    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };
 
    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };
 
    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };
 
    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}
 
use std::cmp::Ordering;
 
/// Equivalent to std::lowerbound and std::upperbound in c++
pub trait BinarySearch<T> {
    fn lower_bound(&self, &T) -> usize;
    fn upper_bound(&self, &T) -> usize;
}
 
impl<T: Ord> BinarySearch<T> for [T] {
    fn lower_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();
 
        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less => {
                    low = mid + 1;
                }
                Ordering::Equal | Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
 
    fn upper_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();
 
        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less | Ordering::Equal => {
                    low = mid + 1;
                }
                Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
}
 
 
 
 
 
 
use std::cmp;
use std::cmp::min;
use std::collections::BTreeMap;
use std::process;
use std::cmp::Ord;
use std::collections::HashMap;
use std::collections::HashSet;
use std::collections::VecDeque;
use std::collections::BTreeSet;
use std::mem;
 
#[derive(PartialEq, Eq)]
pub struct point(i64, i64, i64);
impl Ord for point {
    fn cmp(&self, other: &point) -> Ordering {
        self.0.cmp(&other.0)
    }
}
 
impl PartialOrd for point {
    fn partial_cmp(&self, other: &point) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}
/*
impl PartialEq for point {
    fn eq(&self, other: &point) -> bool {
        self.0 == other.0
    }
}
impl Eq for point {
    fn eq(&self, other: &point) -> bool {
        self.0 == other.0
    }s
}*/
type P = (i64 , i64);
fn bfs(start:&P, goal:&P, map:&Vec<Vec<char>>) -> i64 {
        let dx = vec![1, 0, -1, 0];
        let dy = vec![0, 1, 0, -1];
        let s = (start.0, start.1);
        let g = (goal.0 , goal.1);
        let mut queue:VecDeque<P> = VecDeque::new();
        let mut d:Vec<Vec<i64>> = Vec::new();
        for i in 0..map.len(){
            d.push(Vec::new());
            for j in 0..map[i].len(){
                d[i].push(30000000);
            }
        }
        queue.push_back((s.0, s.1));
        d[s.0 as usize][s.1 as usize] = 0;
        while(queue.len()!=0){
            let mut p:P = queue[0];
            queue.pop_front();
            if p.0 == g.0 && p.1 == g.1{break;}
            for i in 0..4{
                let nx = p.0 + dx[i];
                let ny = p.1 + dy[i];
                if (0<=nx && nx <map.len() as i64 && 0<=ny &&ny<map[0].len() as i64 && map[nx as usize][ny as usize]!='#' && d[nx as usize][ny as usize] == 30000000 ){
                    queue.push_back((nx, ny));
                    d[nx as usize][ny as usize] = d[p.0 as usize][p.1 as usize] + 1;
                }
            }
 
        }
        d[goal.0 as usize][goal.1 as usize]
}
 
fn bfs01(start:&P, goal:&P, map:&Vec<Vec<char>>) -> i64 {
        let dx = vec![1, 0, -1, 0];
        let dy = vec![0, 1, 0, -1];
        let s = (start.0, start.1);
        let g = (goal.0 , goal.1);
        let mut queue:VecDeque<P> = VecDeque::new();
        let mut d:Vec<Vec<i64>> = Vec::new();
        for i in 0..map.len(){
            d.push(Vec::new());
            for j in 0..map[i].len(){
                d[i].push(30000000);
            }
        }
        queue.push_back((s.0, s.1));
        d[s.0 as usize][s.1 as usize] = 0;
        while(queue.len()!=0){
            let mut p:P = queue[0];
            queue.pop_front();
            if p.0 == g.0 && p.1 == g.1{break;}
            for i in 0..4{
                let nx = p.0 + dx[i];
                let ny = p.1 + dy[i];
                if (0<=nx && nx <map.len() as i64 && 0<=ny &&ny<map[0].len() as i64 && d[nx as usize][ny as usize] == 30000000 ){
                    if map[nx as usize][ny as usize] == '#'{
                        queue.push_back((nx, ny));
                        d[nx as usize][ny as usize] = d[p.0 as usize][p.1 as usize] + 1;
                    }
                    else{
                        queue.push_front((nx, ny));
                        d[nx as usize][ny as usize] = d[p.0 as usize][p.1 as usize];
                    }
                }
            }
 
        }
        d[goal.0 as usize][goal.1 as usize]
}
struct UnionFind_W{
    par:Vec<usize>,
    rank:Vec<usize>,
    size:Vec<usize>,
    diff_weight:Vec<i64>,    
}
impl UnionFind_W{
    fn init(n:usize) -> UnionFind_W{
        let mut par = vec![0;n];
        for i in 0..n{
            par[i] = i;    
        }
        UnionFind_W{
            par:par,
            rank:vec![0;n], 
            size:vec![1;n], 
            diff_weight:vec![0i64;n],
        }
    }
    fn find(&mut self, x:usize) ->usize{
        if(self.par[x] == x){
            x    
        }
        else{
            let p = self.par[x];
            let r = self.find(p);
            self.diff_weight[x] += self.diff_weight[p];
            self.par[x] = r;
            r
        }
    }
    fn weight(&mut self,x:usize)->i64{
        self.find(x);
        self.diff_weight[x]
    }
    fn diff(&mut self, x:usize, y:usize)->i64{
        let res = self.weight(x) - self.weight(y);
        res
    }
    fn same(&mut self, a:usize, b:usize)->bool{
        self.find(a) == self.find(b)    
    }
    fn unite(&mut self, xx:usize, yy:usize, ww:i64) -> bool{
        let mut w = ww.clone();
        let mut x = xx.clone();
        let mut y = yy.clone();

        w += self.weight(x);
        w -= self.weight(y);
        x = self.find(x);
        y = self.find(y);
        if (x == y) {return false;}
        if self.rank[x] < self.rank[y]{
            mem::swap(&mut x, &mut y);
            w = -w;
        }

        if self.rank[x] == self.rank[y]{
            self.rank[y] += 1;
        }
        self.par[y] = x;
        self.size[x] += self.size[y];
        self.diff_weight[y] = w;
        return true;
    }
    fn check_size(&mut self, a:usize) -> usize{
        let x = self.find(a);
        let s = self.size[x];
        s    
    }
}





struct UnionFind{
    par:Vec<usize>,
    rank:Vec<usize>,
    size:Vec<usize>,    
}
impl UnionFind{
    fn init(n:usize) -> UnionFind{
        let mut par = vec![0;n];
        for i in 0..n{
            par[i] = i;    
        }
        UnionFind{
            par:par,
            rank:vec![0;n], 
            size:vec![1;n], 
        }
    }
    fn find(&mut self, x:usize) ->usize{
        if(self.par[x] == x){
            x    
        }
        else{
            let p = self.par[x];
            let res = self.find(p);
            self.par[x] = res;
            res
        }
    }
    fn same(&mut self, a:usize, b:usize)->bool{
        self.find(a) == self.find(b)    
    }
    fn unite(&mut self, a:usize, b:usize){
        let x = self.find(a);
        let y = self.find(b);
        if x != y{
            if (self.rank[x] < self.rank[y]){
                self.par[x] = y;
                self.size[y] += self.size[x];
            }
            else{
                self.par[y] = x;
                self.size[x] += self.size[y];
                if(self.rank[x] == self.rank[y]){ self.rank[x]+=1;}
            }
        }
    }
    fn check_size(&mut self, a:usize) -> usize{
        let x = self.find(a);
        let s = self.size[x];
        s    
    }
}
fn gcd(a:usize, b:usize)->usize{
    if b==0{return a;}
    gcd(b, a%b)
}
fn lcm(a: usize, b: usize) -> usize {
    a / gcd(a, b) * b
}
fn extgcd(a:usize, b:usize, x:&mut usize, y:&mut usize)->usize{
    let mut d = a;
    if b != 0{
        d = extgcd(b, a%b, y, x);
        *y -= (a/b) * (*x);    
    }    
    else{
        *x = 1;
        *y = 0;
    }
    d
}

use std::ops::{Deref, DerefMut, Index, IndexMut};
use std::any::{Any, TypeId};

struct vec2<T>(Vec<T>);
 
impl<T> Deref for vec2<T> {
    type Target = Vec<T>;
    fn deref(&self) -> &Vec<T> {
        &self.0
    }
}
 
impl<T> DerefMut for vec2<T> {
    fn deref_mut(&mut self) -> &mut Vec<T> {
        &mut self.0
    }
}
impl<T> Index<i64> for vec2<T> {
    type Output = T;
    fn index(&self, index: i64) -> &T {
        &self.0[index as usize]
    }
}
 
impl<T> IndexMut<i64> for vec2<T> {
    fn index_mut(&mut self, index: i64) -> &mut T {
        &mut self.0[index as usize]
    }
}

fn segment_sieve(a:usize, b:usize) -> Vec<bool>{
    let mut is_prime= vec![false; b+1];
    let mut is_prime_small= vec![false; b+1];
    for i in 0..b{
        if i*i>=b{break;}    
        is_prime_small[i] = true;
    }
    for i in 0..b{
        if i>=b-a{break;}
        is_prime[i] = true;    
    }
    for i in 2..b{
        if i*i>=b{break;}
        if is_prime_small[i] == true{


            let mut j = 2*i;
            while(j*j<b){
                is_prime_small[j] = false;
                j+=i;
            }
            
            j = cmp::max(2, ((a+i-1)/i)*i);
            while(j<b){
                is_prime[j-a] = false;
                j+=i;    
            }    
        }    
    }
    is_prime


}
fn sieve(n:usize) -> (Vec<bool>, Vec<usize>){
    let mut p:usize = 0;
    let mut is_prime = vec![false; n+1];
    let mut prime = Vec::new();
    for i in 0..n+1{
        is_prime[i] = true;
    }
    is_prime[0] = false;
    is_prime[1] = false;
    for i in 2..n+1{
        if is_prime[i]{
            prime.push(i as usize);
            let mut j = 2*i;
            while(j<=n){
                is_prime[j] = false;
                j+=i;
            }
        }
    }
    (is_prime, prime)
    
}
fn divisor(n:usize) -> Vec<usize>{
    let mut res:Vec<usize> = Vec::new(); 
    for i in 1..n+1{
        if i*i>n{break;}
        if n%i == 0{
            res.push(i);
            if i != n/i{
                res.push(n/i);    
            }    
        }    
    }
    res
}

fn prime_factor(nn:usize) -> HashMap<usize, usize>{
    let mut res:HashMap<usize, usize> = HashMap::new();
    let mut n = nn.clone();
    for i in 2..n+1{
        if i*i>n{break;}
        while(n%i == 0){
            *res.entry(i).or_insert(0) += 1;
            n/=i;    
        }    
    }
    if (n != 1){res.insert(n,1);}
    res
}

use std::collections::BinaryHeap;

#[derive(PartialEq)]
struct mf(i64);

impl Eq for mf {}

impl PartialOrd for mf {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }
}

impl Ord for mf {
    fn cmp(&self, other: &mf) -> Ordering {
        self.partial_cmp(other).unwrap()
    }
}




#[derive(Copy, Clone, Eq, PartialEq)]
struct edge1{from:usize, to:usize, cost:i64}
impl Ord for edge1 {
    fn cmp(&self, other: &edge1) -> Ordering {
        other.cost.cmp(&self.cost)
    }
}

impl PartialOrd for edge1 {
    fn partial_cmp(&self, other: &edge1) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}
fn kruskal(es:&mut Vec<edge1>, n:usize) -> i64{
    es.sort();
    es.reverse();
    let mut uf =  UnionFind::init(n);
    let mut res = 0;
    for i in 0..es.len(){
        let e = es[i];
        if uf.same(e.from, e.to) == false{
            uf.unite(e.from, e.to);
            res += e.cost;
        }

    }
    res
}




fn bellman_ford(s:usize, n:usize, inf:i64, es:&Vec<edge1>) -> Vec<i64>{
    let mut d = vec![inf;n];
    d[s] = 0;
    for j in 0..n-1{
        let mut update = false;
        for i in 0..es.len(){
            let e = es[i];
            if (d[e.from] != inf && d[e.to] > d[e.from] + e.cost){
                d[e.to] = d[e.from] + e.cost;
            }
                
        }
    }

    d
}
fn find_negative_loop(n:usize, es:&Vec<edge1>, inf:i64) -> bool{
    let mut d = vec![0;n];
    for i in 0..n{
        for j in 0..es.len(){
            let e = es[j];
            if d[e.to] > d[e.from] + e.cost{
                d[e.to] = d[e.from] + e.cost;
                if (i == n-1) {return false;}
            }
        }
    }
    return true;
}
#[derive(Copy, Clone, Eq, PartialEq)]
struct State{
    cost: i64,
    position: usize,
}
impl Ord for State {
    fn cmp(&self, other: &State) -> Ordering {
        // Notice that the we flip the ordering on costs.
        // In case of a tie we compare positions - this step is necessary
        // to make implementations of `PartialEq` and `Ord` consistent.
        other.cost.cmp(&self.cost)
    }
}

// `PartialOrd` needs to be implemented as well.
impl PartialOrd for State {
    fn partial_cmp(&self, other: &State) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

struct edge2{
    to:usize,
    cost:i64,
}
fn dijkstra(adj_list: &Vec<Vec<edge2>>, start:usize, inf:i64) -> Vec<i64>{
    let mut d= vec![inf;adj_list.len()];
    let mut heap = BinaryHeap::new();
    d[start] = 0;
    heap.push(State{cost: 0, position: start });
    while (heap.len() != 0){
        let p = heap.pop().unwrap();
        let v = p.position;
        if(d[v] < p.cost) {continue;}
        for i in 0..adj_list[v].len(){
            let e = &adj_list[v][i];
            if d[e.to] > d[v] + e.cost{
                d[e.to] = d[v] + e.cost;
                heap.push(State{cost:d[e.to], position:e.to});
            }
        }
            
    }
    d
        
}
// no verify
fn prim(n:usize, cost:&Vec<Vec<i64>>, inf:i64, s:usize) -> i64{
    let mut mincost = vec![inf;n];
    let mut used = vec![false;n];
    mincost[s] = 0;
    let mut res = 0;
    while(true){
        let mut v:i64 = -1;
        for  u in 0..n{
            if (used[u] == false && (v == -1 || mincost[u] <mincost[v as usize])){v = u.clone() as i64;}
        }
        if v == -1{break;}
        used[v as usize] = true;
        res += mincost[v as usize];
        for u in 0.. n{
            mincost[u] = cmp::min(mincost[u], cost[v as usize][u]);
        }
    }
    return res;
}
//
fn find_negative_loop2(n:usize, es:&Vec<edge1>, inf:i64, dist:&mut Vec<i64>) -> Vec<bool>{
    let mut neg = vec![false;n];
    for i in 0..n{
        for j in 0..es.len(){
            
            let e = es[j];
            if dist[e.from] == inf{continue;}

            if dist[e.to] > dist[e.from] + e.cost{
                dist[e.to] = dist[e.from] + e.cost;
                neg[e.to] = true;
            }
            if neg[e.from] == true{ neg[e.to] = true;}
        }
    }
    return neg;
}
fn mod_for_big_int(k:Vec<char>, C:usize)-> usize{
    let mut k_mod_C = 0;
    for i in 0..k.len(){
        k_mod_C = (k_mod_C*10+(k[i] as usize - 48) as usize)%C;
    }
    k_mod_C
}
/*
fn imos() {
    input!{
        n:usize,
        m:usize,
        q:usize,
        lr:[(usize, usize);m],
        pq:[(usize, usize);q],
    }
    let mut v = vec![vec![0;n+2];n+2];
    for i in 0..m{
        v[lr[i].0][lr[i].1] +=1;
    }
    
    for i in 1..n+2{
        for j in 1..n+2{

            v[i][j] +=v[i-1][j];
        }
    }
    for i in 1..n+2{
        for j in 1..n+2{

            v[i][j] +=v[i][j-1];
        }
    }
    for i in 0..q{
        println!("{}", v[pq[i].1][pq[i].1] - v[pq[i].0-1][pq[i].1] - v[pq[i].1][pq[i].0-1] + v[pq[i].0-1][pq[i].0-1] );
    }
}
*/
/*
fn main(){
    input!{
        n:usize,
        xy:[(usize, usize);n-1],
        q:usize,
        ab:[(usize, usize);q],
    }
    let MAX_DEPTH =1+ ( (100000 as f64 ).log2() ) as usize;
    let mut G:Vec<Vec<usize>> = Vec::new();
    let mut root = 1;
    let mut parent = vec![vec![0;n+1];MAX_DEPTH];
    let mut depth = vec![0;n+1];
    fn dfs(G:&Vec<Vec<usize>>,parent:&mut Vec<Vec<i64>>,depth:&mut Vec<i64>, v:i64, p:i64, d:i64){
        let v = v as usize;
        parent[0][v] = p as i64;
        depth[v] = d as i64;
        for i in 0..G[v].len(){
            if G[v][i] != p as usize{
                dfs(G, parent, depth, G[v][i] as i64, v as i64, d+1);
            }
        }
    }
    fn init(G:&Vec<Vec<usize>>, V:usize, root:usize, parent:&mut Vec<Vec<i64>>,depth:&mut Vec<i64>, MAX_DEPTH:usize){
        dfs(G,parent,depth, root as i64, -1, 0);
        for k in 0..(MAX_DEPTH+1){
            if k+1>=MAX_DEPTH{
                break;
            }

            for v in 0..V{
                if parent[k][v] < 0 { parent[k+1][v] = -1;}
                else{ parent[k+1][v] = parent[k][parent[k][v] as usize];}
            }
        }
    }
    fn lca(u:usize, v:usize, parent:& Vec<Vec<i64>>,depth:&Vec<i64>, MAX_DEPTH:usize) -> i64{
        let mut u = u.clone();
        let mut v = v.clone();
        let mut du = depth[u];
        let mut dv = depth[v];
        if depth[u] > depth[v]{mem::swap(&mut u, &mut v);}
        for k in 0..MAX_DEPTH{
            let shifted = (depth[v] - depth[u]) >> k;
            if (shifted & 1) == 1{
                v = parent[k][v] as usize;
            }
            if shifted == 0{
                break;
            }
        }
        //if u==v{return u as i64;}

        if u == v{ return (du-dv).abs() + 1;}
        for k in (0..MAX_DEPTH).rev(){
            if parent[k][u] != parent[k][v]{
                u = parent[k][u] as usize;
                v = parent[k][v] as usize;
            }
        }
        //return parent[0][u];
        let mut dd = 0;
        if parent[0][u] != -1{
            dd = depth[parent[0][u] as usize];
        }
        return du-dd*2+dv+1;
    }
    for i in 0..n+1{
        G.push(Vec::new());
    }
    for i in 0..n-1{
        G[xy[i].0].push(xy[i].1);
        G[xy[i].1].push(xy[i].0);
    }
    let mut parent = vec![vec![0;n+1];MAX_DEPTH];
    let mut depht = vec![0;n+1];
    init(&G, n+1, 1, &mut parent,&mut depth, MAX_DEPTH); 
    for i in 0..q{
        println!("{}", lca(ab[i].0, ab[i].1, &parent, &depth, MAX_DEPTH) );
        }



}
*/
fn main(){
    input!{
        k:usize,
    }
    fn sol(k:usize) -> i64{
        let mut q:VecDeque<(usize, i64)> = VecDeque::new();
        let mut d = vec![100000000000000;k+1];
        q.push_back((1, 1));
        d[1] = 1;
        while(q.len()!=0){
            let mut p = q[0];
            q.pop_front();
            if p.0 == 0{
                break;
            }
            let mut now = p.0;

            
            if d[(now * 10)%k] > d[p.0]{
                q.push_front(( (now*10)%k, p.1));
                d[(now*10%k)] = p.1;
            }
            if d[(now+1)%k] > d[p.0]+1{
                q.push_back(( (now+1)%k, p.1 + 1));
                d[(now+1)%k] = p.1 + 1;

            }
        }
        d[0]
    }
    println!("{}", sol(k));
}