Find the kth smallest number with sum of digits as m

Given two integers M and K, the task is to find the Kth smallest number with digit sum as M.

Examples:

Input: M = 5, K = 3
Output: 23
Sequence of numbers starting from 1 with digit sum as 5 is as follows:
5
14
23
32
41
So 3rd smallest number is 23



Input: M = 4, K = 1
Output: 4

Approach: We need to find the sequence of numbers starting from 1 with the sum of digits as m.

  • Write a recursive function that will increase the numbers until the sum of the digits of the number will be equal to our required sum M.
  • For that, start from 0 always and check for single digit number that will add upto M
  • Let’s take an example for sum = 5 so we will start from 0 and go upto 5 in single digit as 6 exceeds out required sum
  • Now from 5 we will move to two digits number 10 and we will go up to 14 because the sum of digits of 14 is 5 and 15 will exceed the required sum and so on, then further we will move in 20’s and this goes on up to 50 because after 50 till 99 the sum of digits of every number will be greater than 5 so we will skip that.
  • Now we will move in three digits 100, 101, 102, … and similarly this operation will be going on until the sum of digits of the number are equal to 15.
  • We will keep insert the elements into set whose sum of digits is equals to M.
  • Below is the code for recursion:

for(int i=0;i<=min(left, 9LL);i++){
  dfs(num*10+i, left-i, ct+1);  
}

To find kth smallest we need to sort the sequence, so it will be better if we store the numbers in a set in C++, as numbers in a set are arranged in sorted order.

Note that this approach will not work for bigger input values.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
#define int long long
const int N = 2005;
  
set<int> ans;
  
// Recursively moving to add
// all the numbers upto a limit
// with sum of digits as m
void dfs(int num, int left, int ct)
{
    // Max number of digits allowed in
    // a number for this implementation
    if (ct >= 15)
        return;
    if (left == 0)
        ans.insert(num);
    for (int i = 0; i <= min(left, 9LL); i++)
        dfs(num * 10 + i, left - i, ct + 1);
}
  
// Function to return the kth number
// with sum of digits as m
int getKthNum(int m, int k)
{
    dfs(0, m, 0);
  
    int ct = 0;
    for (auto it : ans) {
        ct++;
  
        // The kth smallest number is found
        if (ct == k) {
            return it;
        }
    }
    return -1;
}
  
// Driver code
int main()
{
    int m = 5, k = 3;
  
    cout << getKthNum(m, k);
  
    return 0;
}

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Java

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// Java implementation of the approach
import java.util.*;
  
class GFG 
{
static int N = 2005;
  
static Set<Integer> ans = new LinkedHashSet<Integer>();
  
// Recursively moving to add
// all the numbers upto a limit
// with sum of digits as m
static void dfs(int num, int left, int ct)
{
    // Max number of digits allowed in
    // a number for this implementation
    if (ct >= 15)
        return;
    if (left == 0)
        ans.add(num);
    for (int i = 0; i <= Math.min(left, 9); i++)
        dfs(num * 10 + i, left - i, ct + 1);
}
  
// Function to return the kth number
// with sum of digits as m
static int getKthNum(int m, int k)
{
    dfs(0, m, 0);
  
    int ct = 0;
    for (int it : ans) 
    {
        ct++;
  
        // The kth smallest number is found
        if (ct == k) 
        {
            return it;
        }
    }
    return -1;
}
  
// Driver code
public static void main(String[] args)
{
    int m = 5, k = 3;
  
    System.out.println(getKthNum(m, k));
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation of the approach
  
# define long long
N = 2005
  
ans = dict()
  
# Recursively moving to add
# all the numbers upto a limit
# with sum of digits as m
def dfs(n, left, ct):
  
    # Max nber of digits allowed in
    # a nber for this implementation
    if (ct >= 15):
        return
    if (left == 0):
        ans[n] = 1
    for i in range(min(left, 9) + 1):
        dfs(n * 10 + i, left - i, ct + 1)
  
# Function to return the kth number
# with sum of digits as m
def getKthNum(m, k):
    dfs(0, m, 0)
  
    c = 0
    for it in sorted(ans.keys()):
        c += 1
  
        # The kth smallest number is found
        if (c == k):
            return it
    return -1
  
# Driver code
m = 5
k = 3
  
print(getKthNum(m, k))
  
# This code is contributed by Mohit Kumar

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C#

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// C# implementation of the approach
using System;
using System.Collections.Generic;             
      
class GFG 
{
static int N = 2005;
  
static HashSet<int> ans = new HashSet<int>();
  
// Recursively moving to add
// all the numbers upto a limit
// with sum of digits as m
static void dfs(int num, int left, int ct)
{
    // Max number of digits allowed in
    // a number for this implementation
    if (ct >= 15)
        return;
    if (left == 0)
        ans.Add(num);
    for (int i = 0; 
             i <= Math.Min(left, 9); i++)
        dfs(num * 10 + i, left - i, ct + 1);
}
  
// Function to return the kth number
// with sum of digits as m
static int getKthNum(int m, int k)
{
    dfs(0, m, 0);
  
    int ct = 0;
    foreach (int it in ans) 
    {
        ct++;
  
        // The kth smallest number is found
        if (ct == k) 
        {
            return it;
        }
    }
    return -1;
}
  
// Driver code
public static void Main(String[] args)
{
    int m = 5, k = 3;
  
    Console.WriteLine(getKthNum(m, k));
}
}
  
// This code is contributed by 29AjayKumar

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Output:

23


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