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# 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:
There exists only one pair (1, 9) satisfying
the equation ((1 * 9) + 1 + 9 = 19)

Input: A = 2, B = 8
Output:
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 ``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

 ``

Output:

`10`

Time Complexity: O(b)

Auxiliary Space: O(1)