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Count of numbers between range having only non-zero digits whose sum of digits is N and number is divisible by M

  • Difficulty Level : Hard
  • Last Updated : 11 Aug, 2021

Given a range [L, R] and two positive integers N and M. The task is to count the numbers in the range containing only non-zero digits whose sum of digits is equal to N and the number is divisible by M.
Examples: 
 

Input: L = 1, R = 100, N = 8, M = 2 
Output:
Only 8, 26, 44 and 62 are valid numbers 
Input: L = 1, R = 200, N = 4, M = 11 
Output:
Only 22 and 121 are valid numbers 
 

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Prerequisites : Digit DP
 



Approach: Firstly, if we are able to count the required numbers up to R i.e. in the range [0, R], we can easily reach our answer in the range [L, R] by solving for from zero to R and then subtracting the answer we get after solving for from zero to L – 1. Now, we need to define the DP states. 
DP States
 

  • Since we can consider our number as a sequence of digits, one state is the position at which we are currently in. This position can have values from 0 to 18 if we are dealing with numbers up to 1018. In each recursive call, we try to build the sequence from left to right by placing a digit from 0 to 9.
  • Second state is the sum of the digits we have placed so far.
  • Third state is the remainder which defines the modulus of the number we have made so far modulo M.
  • Another state is the boolean variable tight which tells the number we are trying to build has already become smaller than R so that in the upcoming recursive calls we can place any digit from 0 to 9. If the number has not become smaller, the maximum limit of digit we can place is digit at the current position in R.

For the number to have only non-zero digits, we maintain a variable nonz whose value if 1 tells the first digit in the number we have placed is a non-zero digit and thus, now we can’t place any zero digit in upcoming calls. Otherwise, we can place a zero digit as a leading zero so as to make number of digits in current number smaller than number of digits in upper limit.
Below is the implementation of the above approach: 
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
const int M = 20;
 
// states - position, sum, rem, tight
// sum can have values upto 162, if we
// are dealing with numbers upto 10^18
// when all 18 digits are 9, then sum
// is 18 * 9 = 162
int dp[M][165][M][2];
 
// n is the sum of digits and number should
// be divisible by m
int n, m;
 
// Function to return the count of
// required numbers from 0 to num
int count(int pos, int sum, int rem, int tight,
          int nonz, vector<int> num)
{
    // Last position
    if (pos == num.size()) {
        if (rem == 0 && sum == n)
            return 1;
        return 0;
    }
 
    // If this result is already computed
    // simply return it
    if (dp[pos][sum][rem][tight] != -1)
        return dp[pos][sum][rem][tight];
 
    int ans = 0;
 
    // Maximum limit upto which we can place
    // digit. If tight is 1, means number has
    // already become smaller so we can place
    // any digit, otherwise num[pos]
    int limit = (tight ? 9 : num[pos]);
 
    for (int d = 0; d <= limit; d++) {
 
        // If the current digit is zero
        // and nonz is 1, we can't place it
        if (d == 0 && nonz)
            continue;
        int currSum = sum + d;
        int currRem = (rem * 10 + d) % m;
        int currF = tight || (d < num[pos]);
        ans += count(pos + 1, currSum, currRem,
                     currF, nonz || d, num);
    }
    return dp[pos][sum][rem][tight] = ans;
}
 
// Function to convert x into its digit vector
// and uses count() function to return the
// required count
int solve(int x)
{
    vector<int> num;
    while (x) {
        num.push_back(x % 10);
        x /= 10;
    }
    reverse(num.begin(), num.end());
 
    // Initialize dp
    memset(dp, -1, sizeof(dp));
    return count(0, 0, 0, 0, 0, num);
}
 
// Driver code
int main()
{
    int L = 1, R = 100;
    n = 8, m = 2;
    cout << solve(R) - solve(L);
    return 0;
}

Java




// Java implementation of the approach
import java.util.*;
 
class GFG
{
     
static int M = 20;
 
// states - position, sum, rem, tight
// sum can have values upto 162, if we
// are dealing with numbers upto 10^18
// when all 18 digits are 9, then sum
// is 18 * 9 = 162
static int dp[][][][] = new int [M][165][M][2];
 
// n is the sum of digits and number should
// be divisible by m
static int n, m;
 
// Function to return the count of
// required numbers from 0 to num
static int count(int pos, int sum, int rem, int tight,
        int nonz, Vector<Integer> num)
{
    // Last position
    if (pos == num.size())
    {
        if (rem == 0 && sum == n)
            return 1;
        return 0;
    }
 
    // If this result is already computed
    // simply return it
    if (dp[pos][sum][rem][tight] != -1)
        return dp[pos][sum][rem][tight];
 
    int ans = 0;
 
    // Maximum limit upto which we can place
    // digit. If tight is 1, means number has
    // already become smaller so we can place
    // any digit, otherwise num[pos]
    int limit = (tight != 0 ? 9 : num.get(pos));
 
    for (int d = 0; d <= limit; d++)
    {
 
        // If the current digit is zero
        // and nonz is 1, we can't place it
        if (d == 0 && nonz != 0)
            continue;
        int currSum = sum + d;
        int currRem = (rem * 10 + d) % m;
        int currF = (tight != 0 || (d < num.get(pos))) ? 1 : 0;
        ans += count(pos + 1, currSum, currRem,
                    currF, (nonz != 0 || d != 0) ? 1 : 0, num);
    }
    return dp[pos][sum][rem][tight] = ans;
}
 
// Function to convert x into its digit vector
// and uses count() function to return the
// required count
static int solve(int x)
{
    Vector<Integer> num = new Vector<Integer>();
    while (x != 0)
    {
        num.add(x % 10);
        x /= 10;
    }
    Collections.reverse(num);
 
    // Initialize dp
    for(int i = 0; i < M; i++)
        for(int j = 0; j < 165; j++)
            for(int k = 0; k < M; k++)
                for(int l = 0; l < 2; l++)
                    dp[i][j][k][l]=-1;
     
    return count(0, 0, 0, 0, 0, num);
}
 
// Driver code
public static void main(String args[])
{
    int L = 1, R = 100;
    n = 8; m = 2;
    System.out.print( solve(R) - solve(L));
}
}
 
// This code is contributed by Arnab Kundu

Python3




# Python3 implementation of the approach
 
# Function to return the count of
# required numbers from 0 to num
def count(pos, Sum, rem, tight, nonz, num):
 
    # Last position
    if pos == len(num):
        if rem == 0 and Sum == n:
            return 1
        return 0
 
    # If this result is already computed
    # simply return it
    if dp[pos][Sum][rem][tight] != -1:
        return dp[pos][Sum][rem][tight]
     
    ans = 0
 
    # Maximum limit upto which we can place
    # digit. If tight is 1, means number has
    # already become smaller so we can place
    # any digit, otherwise num[pos]
    if tight:
        limit = 9
    else:
        limit = num[pos]
     
    for d in range(0, limit + 1):
 
        # If the current digit is zero
        # and nonz is 1, we can't place it
        if d == 0 and nonz:
            continue
             
        currSum = Sum + d
        currRem = (rem * 10 + d) % m
        currF = int(tight or (d < num[pos]))
        ans += count(pos + 1, currSum, currRem,
                     currF, nonz or d, num)
     
    dp[pos][Sum][rem][tight] = ans
    return ans
 
# Function to convert x into its digit
# vector and uses count() function to
# return the required count
def solve(x):
 
    num = []
    global dp
     
    while x > 0:
        num.append(x % 10)
        x //= 10
     
    num.reverse()
 
    # Initialize dp
    dp = [[[[-1, -1] for i in range(M)]
                     for j in range(165)]
                     for k in range(M)]
    return count(0, 0, 0, 0, 0, num)
 
# Driver code
if __name__ == "__main__":
 
    L, R = 1, 100
     
    # n is the sum of digits and number
    # should be divisible by m
    n, m, M = 8, 2, 20
     
    # States - position, sum, rem, tight
    # sum can have values upto 162, if we
    # are dealing with numbers upto 10^18
    # when all 18 digits are 9, then sum
    # is 18 * 9 = 162
    dp = []
 
    print(solve(R) - solve(L))
     
# This code is contributed by Rituraj Jain

C#




// C# implementation of the approach
using System;
using System.Collections.Generic;
 
class GFG
{
     
static int M = 20;
 
// states - position, sum, rem, tight
// sum can have values upto 162, if we
// are dealing with numbers upto 10^18
// when all 18 digits are 9, then sum
// is 18 * 9 = 162
static int [,,,]dp = new int [M, 165, M, 2];
 
// n is the sum of digits and number should
// be divisible by m
static int n, m;
 
// Function to return the count of
// required numbers from 0 to num
static int count(int pos, int sum, int rem, int tight,
                            int nonz, List<int> num)
{
    // Last position
    if (pos == num.Count)
    {
        if (rem == 0 && sum == n)
            return 1;
        return 0;
    }
 
    // If this result is already computed
    // simply return it
    if (dp[pos,sum,rem,tight] != -1)
        return dp[pos,sum,rem,tight];
 
    int ans = 0;
 
    // Maximum limit upto which we can place
    // digit. If tight is 1, means number has
    // already become smaller so we can place
    // any digit, otherwise num[pos]
    int limit = (tight != 0 ? 9 : num[pos]);
 
    for (int d = 0; d <= limit; d++)
    {
 
        // If the current digit is zero
        // and nonz is 1, we can't place it
        if (d == 0 && nonz != 0)
            continue;
        int currSum = sum + d;
        int currRem = (rem * 10 + d) % m;
        int currF = (tight != 0 || (d < num[pos])) ? 1 : 0;
        ans += count(pos + 1, currSum, currRem,
                    currF, (nonz != 0 || d != 0) ? 1 : 0, num);
    }
    return dp[pos, sum, rem, tight] = ans;
}
 
// Function to convert x into its digit vector
// and uses count() function to return the
// required count
static int solve(int x)
{
    List<int> num = new List<int>();
    while (x != 0)
    {
        num.Add(x % 10);
        x /= 10;
    }
    num.Reverse();
 
    // Initialize dp
    for(int i = 0; i < M; i++)
        for(int j = 0; j < 165; j++)
            for(int k = 0; k < M; k++)
                for(int l = 0; l < 2; l++)
                    dp[i, j, k, l] = -1;
     
    return count(0, 0, 0, 0, 0, num);
}
 
// Driver code
public static void Main(String []args)
{
    int L = 1, R = 100;
    n = 8; m = 2;
    Console.Write( solve(R) - solve(L));
}
}
 
// This code has been contributed by 29AjayKumar

Javascript




<script>
 
// JavaScript implementation of the approach
 
var M = 20;
 
// states - position, sum, rem, tight
// sum can have values upto 162, if we
// are dealing with numbers upto 10^18
// when all 18 digits are 9, then sum
// is 18 * 9 = 162
var dp = Array.from(Array(M), ()=>Array(165));
 
// n is the sum of digits and number should
// be divisible by m
var n, m;
 
// Function to return the count of
// required numbers from 0 to num
function count( pos, sum, rem, tight, nonz, num)
{
    // Last position
    if (pos == num.length) {
        if (rem == 0 && sum == n)
            return 1;
        return 0;
    }
    // If this result is already computed
    // simply return it
    if (dp[pos][sum][rem][tight] != -1)
        return dp[pos][sum][rem][tight];
 
    var ans = 0;
 
    // Maximum limit upto which we can place
    // digit. If tight is 1, means number has
    // already become smaller so we can place
    // any digit, otherwise num[pos]
    var limit = (tight ? 9 : num[pos]);
 
    for (var d = 0; d <= limit; d++) {
 
        // If the current digit is zero
        // and nonz is 1, we can't place it
        if (d == 0 && nonz)
            continue;
        var currSum = sum + d;
        var currRem = (rem * 10 + d) % m;
        var currF = (tight != 0 || (d < num[pos])) ? 1 : 0;
        ans += count(pos + 1, currSum, currRem,
                    currF, (nonz != 0 || d != 0) ? 1 : 0, num);
    }
    dp[pos][sum][rem][tight] = ans;
    return ans;
}
 
// Function to convert x into its digit vector
// and uses count() function to return the
// required count
function solve(x)
{
    var num = [];
    while (x) {
        num.push(x % 10);
        x = parseInt(x/10);
    }
    num.reverse();
 
    // Initialize dp
    for(var i =0; i<M; i++)
        for(var j =0; j<165; j++)
            dp[i][j] = Array.from(Array(M), ()=>Array(2).fill(-1));
    return count(0, 0, 0, 0, 0, num);
}
 
// Driver code
var L = 1, R = 100;
n = 8, m = 2;
document.write( solve(R) - solve(L));
 
</script>
Output: 
4

 

Time Complexity: O(M*M)
Auxiliary Space: O(M*M)

Short Python Implementation : 
 

Python3




# User Input
l, r, n, m = 1, 100, 8, 2
 
# Initialize Result
output = []
 
# Traverse through all numbers
for x in range(l, r+1):
 
    # Check for all conditions in every number
    if sum([int(k) for k in str(x)]) == n and x % m == 0 and '0' not in str(x): # Check conditions
        output.append(x)
  
print(len(output)) 
  
# This code is contributed by mailprakashindia



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