Count of pairs satisfying the given condition

Given two integers A and B, the task is to calculate the number of pairs (a, b) such that 1 ≤ a ≤ A, 1 ≤ b ≤ B and the equation (a * b) + a + b = concat(a, b) is true where conc(a, b) is the concatenation of a and b (for example, conc(12, 23) = 1223, conc(100, 11) = 10011). Note that a and b should not contain any leading zeroes.

Examples:

Input: A = 1, B = 12 
Output: 1 
There exists only one pair (1, 9) satisfying 
the equation ((1 * 9) + 1 + 9 = 19)

Input: A = 2, B = 8 
Output: 0 
There doesn’t exist any pair satisfying the equation. 

Approach: It can be observed that the above (a * b + a + b = conc(a, b)) will only be satisfied when the digits of an integer ≤ b contains only 9. Simply, calculate the number of digits (≤ b) containing only 9 and multiply with the integer a.

Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function to return the number of
// pairs satisfying the equation
int countPair(int a, int b)
{
    // Converting integer b to string
    // by using to_string function
    string s = to_string(b);

    // Loop to check if all the digits
    // of b are 9 or not
    int i;
    for (i = 0; i < s.length(); i++) {

        // If '9' doesn't appear
        // then break the loop
        if (s[i] != '9')
            break;
    }

    int result;

    // If all the digits of b contain 9
    // then multiply a with string length
    // else multiply a with string length - 1
    if (i == s.length())
        result = a * s.length();
    else
        result = a * (s.length() - 1);

    // Return the number of pairs
    return result;
}

// Driver code
int main()
{
    int a = 5, b = 101;

    cout << countPair(a, b);

    return 0;
}

Java

// Java implementation of the approach
class GFG
{

// Function to return the number of
// pairs satisfying the equation
static int countPair(int a, int b)
{
    // Converting integer b to String
    // by using to_String function
    String s = String.valueOf(b);

    // Loop to check if all the digits
    // of b are 9 or not
    int i;
    for (i = 0; i < s.length(); i++)
    {

        // If '9' doesn't appear
        // then break the loop
        if (s.charAt(i) != '9')
            break;
    }

    int result;

    // If all the digits of b contain 9
    // then multiply a with String length
    // else multiply a with String length - 1
    if (i == s.length())
        result = a * s.length();
    else
        result = a * (s.length() - 1);

    // Return the number of pairs
    return result;
}

// Driver code
public static void main(String[] args)
{
    int a = 5, b = 101;

    System.out.print(countPair(a, b));
}
}

// This code is contributed by PrinciRaj1992

Python3

# Python3 implementation of the approach

# Function to return the number of
# pairs satisfying the equation
def countPair(a, b):

    # Converting integer b to string
    # by using to_function
    s = str(b)

    # Loop to check if all the digits
    # of b are 9 or not
    i = 0
    while i < (len(s)):

        # If '9' doesn't appear
        # then break the loop
        if (s[i] != '9'):
            break
        i += 1

    result = 0

    # If all the digits of b contain 9
    # then multiply a with length
    # else multiply a with length - 1
    if (i == len(s)):
        result = a * len(s)
    else:
        result = a * (len(s) - 1)

    # Return the number of pairs
    return result

# Driver code
a = 5
b = 101

print(countPair(a, b))

# This code is contributed by mohit kumar 29

C#

// C# implementation of the approach
using System;

class GFG
{

// Function to return the number of
// pairs satisfying the equation
static int countPair(int a, int b)
{
    // Converting integer b to String
    // by using to_String function
    String s = String.Join("", b);

    // Loop to check if all the digits
    // of b are 9 or not
    int i;
    for (i = 0; i < s.Length; i++)
    {

        // If '9' doesn't appear
        // then break the loop
        if (s[i] != '9')
            break;
    }

    int result;

    // If all the digits of b contain 9
    // then multiply a with String length
    // else multiply a with String length - 1
    if (i == s.Length)
        result = a * s.Length;
    else
        result = a * (s.Length - 1);

    // Return the number of pairs
    return result;
}

// Driver code
public static void Main(String[] args)
{
    int a = 5, b = 101;

    Console.Write(countPair(a, b));
}
}

// This code is contributed by Rajput-Ji

Javascript

<script>

// Javascript implementation of the approach

// Function to return the number of
// pairs satisfying the equation
function countPair(a, b)
{

    // Converting integer b to string
    // by using to_string function
    var s = (b.toString());

    // Loop to check if all the digits
    // of b are 9 or not
    var i;
    for(i = 0; i < s.length; i++)
    {

        // If '9' doesn't appear
        // then break the loop
        if (s[i] != '9')
            break;
    }

    var result;

    // If all the digits of b contain 9
    // then multiply a with string length
    // else multiply a with string length - 1
    if (i == s.length)
        result = a * s.length;
    else
        result = a * (s.length - 1);

    // Return the number of pairs
    return result;
}

// Driver code
var a = 5, b = 101;
document.write(countPair(a, b));

// This code is contributed by rutvik_56

</script>

Output:

10

Time Complexity: O(b)

Auxiliary Space: O(1)