Open In App

# Find x and y satisfying ax + by = n

Given a, b and n. Find x and y that satisfies ax + by = n. Print any of the x and y satisfying the equation
Examples :

```Input : n=7 a=2 b=3
Output : x=2, y=1
Explanation: here x and y satisfies the equation

Input : 4 2 7
Output : No Solution Exists```

We can check if any solutions exists or not using Linear Diophantine Equations, but here we need to find out the solutions for this equation, we can simply iterate for all possible values from 0 to n as it cannot exceed n for this given equation. So solving this equation with pen and paper gives y=(n-ax)/b and similarly we get the other number to be x=(n-by)/a. But this way we would get only positive solution for x and y may be positive or negative depends on sign of b.

using Linear Diophantine Equation, we can say “no solution”  only when GCD(a, b) would not be a divisor of n. otherwise, solution exists.

## C++14

 `// CPP program to find solution of ax + by = n``#include ``using` `namespace` `std;` `// function to find the solution``int` `gcd(``int` `a, ``int` `b,``int` `&x, ``int` `&y){``    ``if``(b == 0){``        ``x= 1; y = 0;``        ``return` `a;``    ``}``    ``int` `x1, y1;``    ``int` `d = gcd(b, a%b, x1, y1);``    ``x = y1;``    ``y = x1 - y1*(a/b);``    ``return` `d;``}``void` `solution(``int` `a, ``int` `b, ``int` `n)``{``    ``int` `x0, y0;``    ``int` `g = gcd(a, b, x0, y0);``    ``if``(n%g != 0){``        ``cout<<``"No Solution Exists"``<

## Java

 `// Java program to find solution``// of ax + by = n``import` `java.io.*;` `class` `GfG {``        ` `    ``// function to find the solution``    ``static` `void` `solution(``int` `a, ``int` `b, ``int` `n)``    ``{``        ``// traverse for all possible values``        ``for` `(``int` `i = ``0``; i * a <= n; i++)``        ``{``    ` `            ``// check if it is satisfying the equation``            ``if` `((n - (i * a)) % b == ``0``)``            ``{``                ``System.out.println(``"x = "` `+ i +``                                   ``", y = "` `+``                                   ``(n - (i * a)) / b);``                ` `                ``return` `;``            ``}``        ``}``    ` `        ``System.out.println(``"No solution"``);``    ``}``    ` `    ` `    ``public` `static` `void` `main (String[] args)``    ``{``        ``int` `a = ``2``, b = ``3``, n = ``7``;``        ``solution(a, b, n);``    ` `    ``}``}` `// This code is contributed by Gitanjali.`

## Python3

 `# Python3 code to find solution of``# ax + by = n` `# function to find the solution``def` `solution (a, b, n):` `    ``# traverse for all possible values``    ``i ``=` `0``    ``while` `i ``*` `a <``=` `n:``        ` `        ``# check if it is satisfying``        ``# the equation``        ``if` `(n ``-` `(i ``*` `a)) ``%` `b ``=``=` `0``:``            ``print``(``"x = "``,i ,``", y = "``,``               ``int``((n ``-` `(i ``*` `a)) ``/` `b))``            ``return` `0``        ``i ``=` `i ``+` `1``    ` `    ``print``(``"No solution"``)` `# driver program to test the above function``a ``=` `2``b ``=` `3``n ``=` `7``solution(a, b, n)` `# This code is contributed by "Sharad_Bhardwaj".`

## C#

 `// C# program to find solution``// of ax + by = n``using` `System;` `class` `GfG {``        ` `    ``// function to find the solution``    ``static` `void` `solution(``int` `a, ``int` `b, ``int` `n)``    ``{``        ` `        ``// traverse for all possible values``        ``for` `(``int` `i = 0; i * a <= n; i++)``        ``{``    ` `            ``// check if it is satisfying the``            ``// equation``            ``if` `((n - (i * a)) % b == 0)``            ``{``                ``Console.Write(``"x = "` `+ i +``                                ``", y = "` `+``                        ``(n - (i * a)) / b);``                ` `                ``return` `;``            ``}``        ``}``    ` `        ``Console.Write(``"No solution"``);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `Main ()``    ``{``        ``int` `a = 2, b = 3, n = 7;``        ``solution(a, b, n);``    ` `    ``}``}` `// This code is contributed by Vt_m.`

## PHP

 ``

## Javascript

 ``

Output

```x = -7, y = 7
x = -16, y = 13
x = -13, y = 11
x = -10, y = 9
x = -7, y = 7
x = -4, y = 5
x = -1, y = 3
x = 2, y = 1
```

Time Complexity: O(log(min(a, b))), as we are using a Euclid gcd function.
Auxiliary Space: O(1), since no extra space has been taken.

The above written code is not handling base cases.

when n=0, a= 0, b= 0 [ ans = infinite solution ]

when n != 0, a = 0 , b = 0 [ ans = no solution ]

when n = 0, a != 0 , b!=0 [ ans = one solution exists ]