# Pair of integers (a, b) which satisfy the given equations

Given the system of equations a2 + b = n and a + b2 = m. The task is to find the number of pair of positive integers (a, b) which satisfy the equation for given n and m.

Examples:

Input: n = 9, m = 3
Output: 1
Explanation:
Only one pair (3, 0) exists for both equations satisfying the conditions.

Input: n = 4, m = 20
Output: 0
Explanation:
There are no such pair exists.

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

Approach:
The approach is to check for all possible pairs of numbers and check if that pair satisfy both the equations or not. For this we have,

```   a2 + b = n ... (1)
a + b2 = m ... (2)
For equation (2),
=> a = m - b2 ... (3)
```
• Now for the positive value of a, every value of b must be from 0 to sqrt(m).
• Obtain the value of a from equations (3).
• If the pair (a, b) satisfy equation (1), then pair (a, b) is the solution of system of equations.

Below is the implementation of the above approach:

## C++

 `// C++ program to count the pair of integers(a, b) ` `// which satisfy the equation ` `// a^2 + b = n and a + b^2 = m ` `#include ` `using` `namespace` `std; ` ` `  `// Function to count valid pairs ` `int` `pairCount(``int` `n, ``int` `m) ` `{ ` `    ``int` `cnt = 0, b, a; ` `    ``for` `(b = 0; b <= ``sqrt``(m); b++) { ` `        ``a = m - b * b; ` `        ``if` `(a * a + b == n) { ` `            ``cnt++; ` `        ``} ` `    ``} ` `    ``return` `cnt; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 9, m = 3; ` ` `  `    ``cout << pairCount(n, m) << endl; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to count the pair of integers(a, b) ` `// which satisfy the equation ` `// a^2 + b = n and a + b^2 = m ` `class` `GFG ` `{ ` ` `  `// Function to count valid pairs ` `static` `int` `pairCount(``int` `n, ``int` `m) ` `{ ` `    ``int` `cnt = ``0``, b, a; ` `    ``for` `(b = ``0``; b <= Math.sqrt(m); b++)  ` `    ``{ ` `        ``a = m - b * b; ` `        ``if` `(a * a + b == n) ` `        ``{ ` `            ``cnt++; ` `        ``} ` `    ``} ` `    ``return` `cnt; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `n = ``9``, m = ``3``; ` `    ``System.out.print(pairCount(n, m) +``"\n"``); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

## Python3

 `# Python3 program to count the pair of integers(a, b) ` `# which satisfy the equation ` `# a^2 + b = n and a + b^2 = m ` ` `  `# Function to count valid pairs ` `def` `pairCount(n, m): ` `    ``cnt ``=` `0``; ` `    ``for` `b ``in` `range``(``int``(``pow``(m, ``1``/``2``))): ` `        ``a ``=` `m ``-` `b ``*` `b; ` `        ``if` `(a ``*` `a ``+` `b ``=``=` `n): ` `            ``cnt ``+``=` `1``; ` `         `  `    ``return` `cnt; ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``n ``=` `9``; ` `    ``m ``=` `3``; ` `    ``print``(pairCount(n, m)); ` `     `  `# This code is contributed by 29AjayKumar `

## C#

 `// C# program to count the pair of integers(a, b) ` `// which satisfy the equation ` `// a^2 + b = n and a + b^2 = m ` `using` `System; ` ` `  `class` `GFG ` `{ ` ` `  `// Function to count valid pairs ` `static` `int` `pairCount(``int` `n, ``int` `m) ` `{ ` `    ``int` `cnt = 0, b, a; ` `    ``for` `(b = 0; b <= Math.Sqrt(m); b++)  ` `    ``{ ` `        ``a = m - b * b; ` `        ``if` `(a * a + b == n) ` `        ``{ ` `            ``cnt++; ` `        ``} ` `    ``} ` `    ``return` `cnt; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `n = 9, m = 3; ` `    ``Console.Write(pairCount(n, m) +``"\n"``); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:

```1
```

Time Complexity: O(sqrt(min(n,m))

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