Count of integers obtained by replacing ? in the given string that give remainder 5 when divided by 13
Given a string str of length N. The task is to find the number of integers obtained by replacing ‘?’ with any digit such that the formed integer gives remainder 5 when it is divided by 13.
Numbers can also begin with zero. The answer can be very large, so, output answer modulo 109 + 7.
Examples:
Input: str = “?44”
Output: 1
Only possible number is 044Input: str = “7?4”
Output: 0Input: str = “8?3?4233?4?”
Output: 770
Approach: Let dp[i][j] be the number of ways to create an i-digit number consistent with the first i digits of the given pattern and congruent to j modulo 13. As our base case, dp[0][i]=0 for i from 1 to 12, and dp[0][0]=1 (as our length-zero number has value zero and thus is zero mod 13.)
Notice that appending a digit k to the end of a number that’s j mod 13 gives a number that’s congruent to 10j+k mod 13. We use this fact to perform our transitions. For every state, dp[i][j] with i < N, iterate over the possible values of k. (If s[i]=’?’, there will be ten choices for k, and otherwise, there will only be one choice.) Then, we add dp[i][j] to dp[i+1][(10j+k)%13].
To get our final answer, we can simply print dp[N][5].
Below is the implementation of the above approach :
C++
// C++ implementation of the approach #include <bits/stdc++.h> using namespace std; #define MOD (int)(1e9 + 7) // Function to find the count of integers // obtained by replacing '?' in a given // string such that formed integer // gives remainder 5 when it is divided by 13 int modulo_13(string s, int n) { long long dp[n + 1][13] = { { 0 } }; // Initialise dp[0][0] = 1; for (int i = 0; i < n; i++) { for (int j = 0; j < 10; j++) { int nxt = s[i] - '0'; // Place digit j at ? position if (s[i] == '?') nxt = j; // Get the remainder for (int k = 0; k < 13; k++) { int rem = (10 * k + nxt) % 13; dp[i + 1][rem] += dp[i][k]; dp[i + 1][rem] %= MOD; } if (s[i] != '?') break; } } // Return the required answer return (int)dp[n][5]; } // Driver code int main() { string s = "?44"; int n = s.size(); cout << modulo_13(s, n); return 0; } |
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Java
// Java implementation of the approach class GFG { static int MOD = (int)(1e9 + 7); // Function to find the count of integers // obtained by replacing '?' in a given // string such that formed integer // gives remainder 5 when it is divided by 13 static int modulo_13(String s, int n) { long [][]dp = new long[n + 1][13]; // Initialise dp[0][0] = 1; for (int i = 0; i < n; i++) { for (int j = 0; j < 10; j++) { int nxt = s.charAt(i) - '0'; // Place digit j at ? position if (s.charAt(i) == '?') nxt = j; // Get the remainder for (int k = 0; k < 13; k++) { int rem = (10 * k + nxt) % 13; dp[i + 1][rem] += dp[i][k]; dp[i + 1][rem] %= MOD; } if (s.charAt(i) != '?') break; } } // Return the required answer return (int)dp[n][5]; } // Driver code public static void main(String []args) { String s = "?44"; int n = s.length(); System.out.println(modulo_13(s, n)); } } // This code is contributed by Rajput-Ji |
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Python3
# Python3 implementation of the approach import numpy as np MOD = (int)(1e9 + 7) # Function to find the count of integers # obtained by replacing '?' in a given # string such that formed integer # gives remainder 5 when it is divided by 13 def modulo_13(s, n) : dp = np.zeros((n + 1, 13)); # Initialise dp[0][0] = 1; for i in range(n) : for j in range(10) : nxt = ord(s[i]) - ord('0'); # Place digit j at ? position if (s[i] == '?') : nxt = j; # Get the remainder for k in range(13) : rem = (10 * k + nxt) % 13; dp[i + 1][rem] += dp[i][k]; dp[i + 1][rem] %= MOD; if (s[i] != '?') : break; # Return the required answer return int(dp[n][5]); # Driver code if __name__ == "__main__" : s = "?44"; n = len(s); print(modulo_13(s, n)); # This code is contributed by AnkitRai01 |
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C#
// C# implementation of the approach using System; class GFG { static int MOD = (int)(1e9 + 7); // Function to find the count of integers // obtained by replacing '?' in a given // string such that formed integer // gives remainder 5 when it is divided by 13 static int modulo_13(String s, int n) { long [,]dp = new long[n + 1, 13]; // Initialise dp[0, 0] = 1; for (int i = 0; i < n; i++) { for (int j = 0; j < 10; j++) { int nxt = s[i] - '0'; // Place digit j at ? position if (s[i] == '?') nxt = j; // Get the remainder for (int k = 0; k < 13; k++) { int rem = (10 * k + nxt) % 13; dp[i + 1, rem] += dp[i, k]; dp[i + 1, rem] %= MOD; } if (s[i] != '?') break; } } // Return the required answer return (int)dp[n,5]; } // Driver code public static void Main(String []args) { String s = "?44"; int n = s.Length; Console.WriteLine(modulo_13(s, n)); } } // This code is contributed by 29AjayKumar |
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Output:
1
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