Count N-digits numbers made up of even and prime digits at odd and even positions respectively

Given a positive integer N, the task is to find the number of integers of N digits having even digits at odd indices and prime digits at even indices.

Examples:

Input: N = 2
Output: 20
Explanation:
Following are the possible number of 2-digits satisfying the given criteria {20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 50, 52, 54, 56, 58, 70, 72, 74, 76, 78}. Therefore, the count of such number is 20.

Input: N = 5
Output: 1600

Approach: The given problem can be solved using the concept of Permutations and Combinations by observing the fact that there are only 4 choices for the even positions as [2, 3, 5, 7] and 5 choices for the odd positions as [0, 2, 4, 6, 8]. Therefore, the count of N-digits numbers satisfying the given criteria is given by:

total count = 4^P5^Q, where P and Q is the number of even and odd positions respectively.

Efficient Approach:

• Define a variable m with value 1000000007.
• Define a function named “power” which will accept two integer parameters named “x” and “y“. This function will return the value of x^y.
  Inside the function power, initialize a variable res with value 1.
  • Run a loop until y is greater than 0. Inside the loop:
                a. If the last bit of y is 1 (i.e., if y is odd), then multiply res with x and take modulo with m.
                b. Divide y by 2.
                c. Multiply x by x and take modulo with m.
  • Return the value of res.
• Define another function named “countNDigitNumber” which will accept an integer parameter named “N“. This function will return the number of N-digit integers satisfying the given criteria.
  • Inside the function countNDigitNumber, define two integer variables named “ne” and “no“. ne will hold the count of even positions and no will hold the count of odd positions. Initialize ne with N/2 + N%2 and no with floor(N/2).
  • Return the product of power(4, ne) and power(5, no) modulo m.
• Inside the main function:
              a. Define an integer variable N and initialize it with 5.
              b. Call the function countNDigitNumber with parameter N and print the result modulo m.

Below is the implementation of the above approach:

C++

// C++ program for the above approache
#include<bits/stdc++.h>
using namespace std;

int m = 1000000007;

// Function to find the value of x ^ y
int power(int x, int y)
{

    // Stores the value of x ^ y
    int res = 1;

    // Iterate until y is positive
    while (y > 0)
    {

        // If y is odd
        if ((y & 1) != 0)
            res = (res * x) % m;

        // Divide y by 2
        y = y >> 1;

        x = (x * x) % m;
    }

    // Return the value of x ^ y
    return res;
}

// Function to find the number of N-digit
// integers satisfying the given criteria
int countNDigitNumber(int N)
{

    // Count of even positions
    int ne = N / 2 + N % 2;

    // Count of odd positions
    int no = floor(N / 2);

    // Return the resultant count
    return power(4, ne) * power(5, no);
}

// Driver Code
int main()
{
    int N = 5;
    cout << countNDigitNumber(N) % m << endl;
}

// This code is contributed by SURENDRA_GANGWAR

Java

// Java program for the above approach
import java.io.*;
class GFG {

static int m = 1000000007;

// Function to find the value of x ^ y
static int power(int x, int y)
{

    // Stores the value of x ^ y
    int res = 1;

    // Iterate until y is positive
    while (y > 0)
    {

        // If y is odd
        if ((y & 1) != 0)
            res = (res * x) % m;

        // Divide y by 2
        y = y >> 1;

        x = (x * x) % m;
    }

    // Return the value of x ^ y
    return res;
}

// Function to find the number of N-digit
// integers satisfying the given criteria
static int countNDigitNumber(int N)
{

    // Count of even positions
    int ne = N / 2 + N % 2;

    // Count of odd positions
    int no = (int)Math.floor(N / 2);

    // Return the resultant count
    return power(4, ne) * power(5, no);
}

// Driver Code
public static void main(String[] args)
{
    int N = 5;
    System.out.println(countNDigitNumber(N) % m);
}
}

// This code is contributed by sanjoy_62.

Python3

# Python program for the above approach

import math
m = 10**9 + 7

# Function to find the value of x ^ y
def power(x, y):

    # Stores the value of x ^ y
    res = 1

    # Iterate until y is positive
    while y > 0:

        # If y is odd
        if (y & 1) != 0:
            res = (res * x) % m

        # Divide y by 2
        y = y >> 1

        x = (x * x) % m

    # Return the value of x ^ y
    return res

# Function to find the number of N-digit
# integers satisfying the given criteria
def countNDigitNumber(n: int) -> None:

    # Count of even positions
    ne = N // 2 + N % 2

    # Count of odd positions
    no = N // 2

    # Return the resultant count
    return power(4, ne) * power(5, no)

# Driver Code
if __name__ == '__main__':

    N = 5
    print(countNDigitNumber(N) % m)

C#

// C# program for the above approach
using System;

class GFG{

static int m = 1000000007;

// Function to find the value of x ^ y
static int power(int x, int y)
{

    // Stores the value of x ^ y
    int res = 1;

    // Iterate until y is positive
    while (y > 0)
    {

        // If y is odd
        if ((y & 1) != 0)
            res = (res * x) % m;

        // Divide y by 2
        y = y >> 1;

        x = (x * x) % m;
    }

    // Return the value of x ^ y
    return res;
}

// Function to find the number of N-digit
// integers satisfying the given criteria
static int countNDigitNumber(int N)
{

    // Count of even positions
    int ne = N / 2 + N % 2;

    // Count of odd positions
    int no = (int)Math.Floor((double)N / 2);

    // Return the resultant count
    return power(4, ne) * power(5, no);
}


// Driver Code
public static void Main()
{
    int N = 5;
    Console.Write(countNDigitNumber(N) % m);
}
}

// This code is contributed by splevel62.

Javascript

   <script>

        // JavaScript program for the above approache


        var m = 10 ** 9 + 7

        // Function to find the value of x ^ y
        function power(x, y) {

            // Stores the value of x ^ y
            var res = 1

            // Iterate until y is positive
            while (y > 0) {

                // If y is odd
                if ((y & 1) != 0)
                    res = (res * x) % m

                // Divide y by 2
                y = y >> 1

                x = (x * x) % m
            }
            // Return the value of x ^ y
            return res
        }
        // Function to find the number of N-digit
        // integers satisfying the given criteria
        function countNDigitNumber(N) {

            // Count of even positions
            var ne = Math.floor(N / 2) + N % 2

            // Count of odd positions
            var no = Math.floor(N / 2)

            // Return the resultant count
            return power(4, ne) * power(5, no)
        }
        // Driver Code


        let N = 5
        document.write(countNDigitNumber(N) % m);


// This code is contributed by Potta Lokesh
    </script>

Output

1600

Time Complexity: O(log N)
Auxiliary Space: O(1)

Dynamic Approach:

• Define a function “isPrime” that takes an integer “num” as input and checks if it is prime. It returns “true” if the number is prime and “false” otherwise.
• Define a function “countIntegers” that takes an integer “N” as input and returns the count of integers satisfying the given criteria.
  • Create a 2D vector “dp” of size (N+1) x 2 to store the intermediate results. Each element “dp[i][j]” represents the count of integers with “i” digits, where “j=0” indicates even digits at odd indices, and “j=1” indicates prime digits at even indices.
  • Initialize the base cases: “dp[1][0] = 5” and “dp[1][1] = 4” since there are 5 even digits (0, 2, 4, 6, 8) and 4 prime digits (2, 3, 5, 7) for a single-digit number.
  • Iterate from “i = 2” to “N”:
    Calculate the count of even digits at odd indices for “i” digits: “dp[i][0] = 5 * dp[i – 1][1]”.
    Calculate the count of prime digits at even indices for “i” digits: “dp[i][1] = 4 * dp[i – 1][0]”.
  • Return the total count of valid integers: “dp[N][1].

Below is the implementation of the above approach:

C++

#include <iostream>
#include <vector>

using namespace std;

// Function to check if a number is prime
bool isPrime(int num) {
    if (num <= 1)
        return false;
    for (int i = 2; i * i <= num; i++) {
        if (num % i == 0)
            return false;
    }
    return true;
}

// Function to count the number of integers
int countIntegers(int N) {
    vector<vector<int>> dp(N + 1, vector<int>(2));

    // Initialize the base cases
    dp[1][0] = 5;  // Even digits at odd indices
    dp[1][1] = 4;  // Prime digits at even indices

    for (int i = 2; i <= N; i++) {
        // Calculate the count of even digits at odd indices
        dp[i][0] = 5 * dp[i - 1][1];

        // Calculate the count of prime digits at even indices
        dp[i][1] = 4 * dp[i - 1][0];
    }

    // Return the total count of valid integers
    return dp[N][1];
}

int main() {
    int N = 5;

    int count = countIntegers(N);
    cout <<  count <<endl;

    return 0;
}
// This code is contributed by rudra1807raj

Java

import java.util.*;

class Main {
    // Function to check if a number is prime
    static boolean isPrime(int num) {
        if (num <= 1)
            return false;
        for (int i = 2; i * i <= num; i++) {
            if (num % i == 0)
                return false;
        }
        return true;
    }

    // Function to count the number of integers
    static int countIntegers(int N) {
        int[][] dp = new int[N + 1][2];

        // Initialize the base cases
        dp[1][0] = 5;  // Even digits at odd indices
        dp[1][1] = 4;  // Prime digits at even indices

        for (int i = 2; i <= N; i++) {
            // Calculate the count of even digits at odd indices
            dp[i][0] = 5 * dp[i - 1][1];

            // Calculate the count of prime digits at even indices
            dp[i][1] = 4 * dp[i - 1][0];
        }

        // Return the total count of valid integers
        return dp[N][1];
    }

    public static void main(String[] args) {
        int N = 5;

        int count = countIntegers(N);
        System.out.println(count);

        // This Code Is Contributed By Shubham Tiwari.
    }
}

Python3

def isPrime(num):
    if num <= 1:
        return False
    for i in range(2, int(num**0.5) + 1):
        if num % i == 0:
            return False
    return True

def countIntegers(N):
    dp = [[0 for _ in range(2)] for _ in range(N + 1)]

    # Initialize the base cases
    dp[1][0] = 5  # Even digits at odd indices
    dp[1][1] = 4  # Prime digits at even indices

    for i in range(2, N + 1):
        # Calculate the count of even digits at odd indices
        dp[i][0] = 5 * dp[i - 1][1]

        # Calculate the count of prime digits at even indices
        dp[i][1] = 4 * dp[i - 1][0]

    # Return the total count of valid integers
    return dp[N][1]

if __name__ == "__main__":
    N = 5
    count = countIntegers(N)
    print(count)

# This Code Is Contributed By Shubham Tiwari

C#

using System;
using System.Collections.Generic;

class Program {
    // Function to check if a number is prime
    static bool IsPrime(int num) {
        if (num <= 1)
            return false;
        for (int i = 2; i * i <= num; i++) {
            if (num % i == 0)
                return false;
        }
        return true;
    }

    // Function to count the number of integers
    static int CountIntegers(int N) {
        int[][] dp = new int[N + 1][];
        for (int i = 0; i <= N; i++) {
            dp[i] = new int[2];
        }

        // Initialize the base cases
        dp[1][0] = 5;  // Even digits at odd indices
        dp[1][1] = 4;  // Prime digits at even indices

        for (int i = 2; i <= N; i++) {
            // Calculate the count of even digits at odd indices
            dp[i][0] = 5 * dp[i - 1][1];

            // Calculate the count of prime digits at even indices
            dp[i][1] = 4 * dp[i - 1][0];
        }

        // Return the total count of valid integers
        return dp[N][1];
    }

    static void Main() {
        int N = 5;
        int count = CountIntegers(N);
        Console.WriteLine(count);

        // This Code Is Contributed By Shubham Tiwari
    }
}

Javascript

// Function to check if a number is prime
function isPrime(num) {
    if (num <= 1)
        return false;
    for (let i = 2; i * i <= num; i++) {
        if (num % i === 0)
            return false;
    }
    return true;
}

// Function to count the number of integers
function countIntegers(N) {
    let dp = Array(N + 1).fill().map(() => Array(2).fill(0));

    // Initialize the base cases
    dp[1][0] = 5;  // Even digits at odd indices
    dp[1][1] = 4;  // Prime digits at even indices

    for (let i = 2; i <= N; i++) {
        // Calculate the count of even digits at odd indices
        dp[i][0] = 5 * dp[i - 1][1];

        // Calculate the count of prime digits at even indices
        dp[i][1] = 4 * dp[i - 1][0];
    }

    // Return the total count of valid integers
    return dp[N][1];
}

let N = 5;
let count = countIntegers(N);
console.log(count);
//This Code Is Contributed By Shubham Tiwari

Output

1600

Time Complexity: O(N).
Auxiliary Space: O(N).