# Count of Numbers in a Range divisible by m and having digit d in even positions

Given a range represented by two positive integers l and r and two integers d and m. Find the count of numbers lying in the range which is divisible by m and have digit d at even positions of the number. (i.e. digit d should not occur on odd position). Note: Both numbers l and r have same number of digits.

Examples:
Input : l = 10, r = 99, d = 8, m = 2
Output : 8
Explanation :Valid numbers are 18, 28, 38, 48, 58, 68, 78 and 98.
88 is not a valid number since 8 is also present at odd position.

Input : l = 1000, r = 9999, d = 7, m = 19
Output : 6

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 the 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 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.

If the current position is an even position, we simply place digit d and recursively solve for the next positions. But if the current position is an odd position we can place any digit except d and solve for the next positions.

Below is the implementation of the above approach.

## C++

 // CPP Program to find the count of // numbers in a range divisible by m // having digit d at even positions #include using namespace std;    const int M = 20;    // states - position, rem, tight int dp[M][M][2];    // d is required digit and number should // be divisible by m int d, m;    // This function returns the count of // required numbers from 0 to num int count(int pos, int rem, int tight,           vector num) {     // Last position     if (pos == num.size()) {         if (rem == 0)             return 1;         return 0;     }        // If this result is already computed     // simply return it     if (dp[pos][rem][tight] != -1)         return dp[pos][rem][tight];        // If the current position is even, place     // digit d, but since we have considered     // 0-indexing, check for odd positions     if (pos % 2) {         if (tight == 0 && d > num[pos])             return 0;            int currTight = tight;            // At this position, number becomes         // smaller         if (d < num[pos])             currTight = 1;            int res = count(pos + 1, (10 * rem + d)                                      % m,                         currTight, num);         return dp[pos][rem][tight] = res;     }        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 dig = 0; dig <= limit; dig++) {            if (dig == d)             continue;            int currTight = tight;            // At this position, number becomes         // smaller         if (dig < num[pos])             currTight = 1;            // Next recursive call, also set nonz         // to 1 if current digit is non zero         ans += count(pos + 1, (10 * rem + dig)                                   % m,                      currTight, num);     }     return dp[pos][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 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, num); }    // Driver Code to test above functions int main() {     int L = 10, R = 99;     d = 8, m = 2;     cout << solve(R) - solve(L) << endl;        return 0; }

## Java

 // Java Program to find the count of  // numbers in a range divisible by m  // having digit d at even positions    import java.util.*;    class GFG  {        static int M = 20;        // states - position, rem, tight     static Integer[][][] dp = new Integer[M][M][2];        // d is required digit and number should     // be divisible by m     static int d, m;        // This function returns the count of     // required numbers from 0 to num     static int count(int pos, int rem, int tight,                             Vector num)     {            // Last position         if (pos == num.size())          {             if (rem == 0)                 return 1;             return 0;         }            // If this result is already computed         // simply return it         if (dp[pos][rem][tight] != -1)             return dp[pos][rem][tight];            // If the current position is even, place         // digit d, but since we have considered         // 0-indexing, check for odd positions         if (pos % 2 == 1)         {             if (tight == 0 && d > num.elementAt(pos))                 return 0;                int currTight = tight;                // At this position, number becomes             // smaller             if (d < num.elementAt(pos))                 currTight = 1;                int res = count(pos + 1, (10 * rem + d) % m,                                         currTight, num);             return dp[pos][rem][tight] = res;         }            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.elementAt(pos);         for (int dig = 0; dig <= limit; dig++)          {                if (dig == d)                 continue;                int currTight = tight;                // At this position, number becomes             // smaller             if (dig < num.elementAt(pos))                 currTight = 1;                // Next recursive call, also set nonz             // to 1 if current digit is non zero             ans += count(pos + 1, (10 * rem + dig) % m,                                         currTight, num);         }         return dp[pos][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 num = new Vector<>();         while (x > 0)         {             num.add(x % 10);             x /= 10;         }         Collections.reverse(num);            // Initialize dp         for (int i = 0; i < dp.length; i++)             for (int j = 0; j < dp[i].length; j++)                 for (int k = 0; k < dp[i][j].length; k++)                     dp[i][j][k] = -1;            return count(0, 0, 0, num);     }        // Driver Code     public static void main(String[] args)     {         int L = 10, R = 99;         d = 8;         m = 2;         System.out.println(solve(R) - solve(L));     } }    // This code is contributed by // sanjeev2552

## Python3

 # Python3 Program to find the count of  # numbers in a range divisible by m  # having digit d at even positions     # This Function returns the count of  # required numbers from 0 to num  def count(pos, rem, tight, num):         # Last position      if pos == len(num):          if rem == 0:              return 1         return 0            # If this result is already     # computed simply return it      if dp[pos][rem][tight] != -1:          return dp[pos][rem][tight]         # If the current position is even,      # place digit d, but since we have      # considered 0-indexing, check for      # odd positions      if pos % 2 == 1:         if tight == 0 and d > num[pos]:              return 0            currTight = tight             # At this position, number          # becomes smaller          if d < num[pos]:              currTight = 1            res = count(pos + 1, (10 * rem + d) % m,                                   currTight, num)                    dp[pos][rem][tight] = res                  return res            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]      limit = 9 if tight else num[pos]         for dig in range(0, limit + 1):          if dig == d:             continue            currTight = tight             # At this position, number becomes          # smaller          if dig < num[pos]:              currTight = 1            # Next recursive call, also set nonz          # to 1 if current digit is non zero          ans += count(pos + 1, (10 * rem + dig) % m,                                      currTight, num)             dp[pos][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):             global dp     num = []      while x > 0:          num.append(x % 10)          x = x // 10            num.reverse()      # Initialize dp with -1     dp = [[[-1, -1] for x in range(M)]                      for y in range(M)]            return count(0, 0, 0, num)     # Driver Code if __name__ == "__main__":        L, R = 10, 99            # d is required digit and number     # should be divisible by m      d, m = 8, 2     M = 20            # states - position, rem, tight     dp = []     print(solve(R) - solve(L))    # This code is contributed  # by Rituraj Jain

## C#

 // C# Program to find the count of  // numbers in a range divisible by m  // having digit d at even positions using System; using System.Collections.Generic;    class GFG  {        static int M = 20;        // states - position, rem, tight     static int[,,] dp = new int[M, M, 2];        // d is required digit and number should     // be divisible by m     static int d, m;        // This function returns the count of     // required numbers from 0 to num     static int count(int pos, int rem, int tight,                             List num)     {            // Last position         if (pos == num.Count)          {             if (rem == 0)                 return 1;             return 0;         }            // If this result is already computed         // simply return it         if (dp[pos, rem, tight] != -1)             return dp[pos, rem, tight];            // If the current position is even, place         // digit d, but since we have considered         // 0-indexing, check for odd positions         if (pos % 2 == 1)         {             if (tight == 0 && d > num[pos])                 return 0;                int currTight = tight;                // At this position, number becomes             // smaller             if (d < num[pos])                 currTight = 1;                int res = count(pos + 1, (10 * rem + d) % m,                                         currTight, num);             return dp[pos, rem, tight] = res;         }            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 dig = 0; dig <= limit; dig++)          {                if (dig == d)                 continue;                int currTight = tight;                // At this position, number becomes             // smaller             if (dig < num[pos])                 currTight = 1;                // Next recursive call, also set nonz             // to 1 if current digit is non zero             ans += count(pos + 1, (10 * rem + dig) % m,                                         currTight, num);         }         return dp[pos, 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 num = new List();         while (x > 0)         {             num.Add(x % 10);             x /= 10;         }         num.Reverse();            // Initialize dp         for (int i = 0; i < dp.GetLength(0); i++)             for (int j = 0; j < dp.GetLength(1); j++)                 for (int k = 0; k < dp.GetLength(2); k++)                     dp[i, j, k] = -1;            return count(0, 0, 0, num);     }        // Driver Code     public static void Main(String[] args)     {         int L = 10, R = 99;         d = 8;         m = 2;         Console.WriteLine(solve(R) - solve(L));     } }    // This code is contributed by Rajput-Ji

## PHP

 \$num[\$pos])              return 0;             \$currTight = \$tight;             // At this position, number becomes          // smaller          if (\$GLOBALS['d'] < \$num[\$pos])              \$currTight = 1;             \$res = count_num(\$pos + 1, (10 * \$rem +                           \$GLOBALS['d']) % \$GLOBALS['m'],                           \$currTight, \$num);          return \$dp[\$pos][\$rem][\$tight] = \$res;      }         \$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]      \$limit = (\$tight ? 9 : \$num[\$pos]);         for (\$dig = 0; \$dig <= \$limit; \$dig++)     {             if (\$dig == \$GLOBALS['d'])              continue;             \$currTight = \$tight;             // At this position, number becomes          // smaller          if (\$dig < \$num[\$pos])              \$currTight = 1;             // Next recursive call, also set nonz          // to 1 if current digit is non zero          \$ans += count_num(\$pos + 1, (10 * \$rem + \$dig) %                            \$GLOBALS['m'], \$currTight, \$num);      }      return \$dp[\$pos][\$rem][\$tight] = \$ans;  }     // Function to convert x into its digit  // vector and uses count() function to  // return the required count  function solve(\$x)  {      \$num = array() ;     while (\$x)      {          array_push(\$num, \$x % 10);          \$x = floor(\$x / 10);      }      \$num = array_reverse(\$num) ;        // Initialize dp      for(\$i = 0 ; \$i < \$GLOBALS['M'] ; \$i++)         for(\$j = 0; \$j < \$GLOBALS['M']; \$j++)             for(\$k = 0; \$k < 2; \$k ++)                 \$GLOBALS['dp'][\$i][\$j][\$k] = -1;                    return count_num(0, 0, 0, \$num);  }     // Driver Code \$GLOBALS['M'] = 20;     // states - position, rem, tight  \$GLOBALS['dp'] = array(array(array()));    \$L = 10; \$R = 99;     // d is required digit and number  // should be divisible by m  \$GLOBALS['d'] = 8 ; \$GLOBALS['m'] = 2;     echo solve(\$R) - solve(\$L) ;     // This code is contributed by Ryuga ?>

Output:

8

Time Complexity : O(18 * (m – 1) * 2), if we are dealing with the numbers upto 1018

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