# Find smallest values of x and y such that ax – by = 0

Given two values ‘a’ and ‘b’ that represent coefficients in “ax – by = 0”, find the smallest values of x and y that satisfy the equation. It may also be assumed that x > 0, y > 0, a > 0 and b > 0.

Input: a = 25, b = 35 Output: x = 7, y = 5

A **Simple Solution** is to try every possible value of x and y starting from 1, 1 and stop when the equation is satisfied.

A **Direct Solution** is to use Least Common Multiple (LCM). LCM of ‘a’ and ‘b’ represents the smallest value that can make both sides equal. We can find LCM using below formula.

LCM(a, b) = (a * b) / GCD(a, b)

Greatest Common Divisor (GCD) can be computed using Euclid’s algorithm.

## C++

`// C++ program to find the smallest values of x and y that ` `// satisfy "ax - by = 0" ` `#include <iostream> ` `using` `namespace` `std; ` ` ` `// To find GCD using Eculcid's algorithm ` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(b == 0) ` ` ` `return` `a; ` ` ` `return` `(gcd(b, a % b)); ` `} ` ` ` `// Prints smallest values of x and y that ` `// satisfy "ax - by = 0" ` `void` `findSmallest(` `int` `a, ` `int` `b) ` `{ ` ` ` `// Find LCM ` ` ` `int` `lcm = (a * b) / gcd(a, b); ` ` ` ` ` `cout << ` `"x = "` `<< lcm / a ` ` ` `<< ` `"\ny = "` `<< lcm / b; ` `} ` ` ` `// Driver program ` `int` `main() ` `{ ` ` ` `int` `a = 25, b = 35; ` ` ` `findSmallest(a, b); ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find the smallest values of ` `// x and y that satisfy "ax - by = 0" ` `class` `GFG { ` ` ` ` ` `// To find GCD using Eculcid's algorithm ` ` ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` ` ` `if` `(b == ` `0` `) ` ` ` `return` `a; ` ` ` `return` `(gcd(b, a % b)); ` ` ` `} ` ` ` ` ` `// Prints smallest values of x and y that ` ` ` `// satisfy "ax - by = 0" ` ` ` `static` `void` `findSmallest(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` ` ` `// Find LCM ` ` ` `int` `lcm = (a * b) / gcd(a, b); ` ` ` ` ` `System.out.print(` `"x = "` `+ lcm / a ` ` ` `+ ` `"\ny = "` `+ lcm / b); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{ ` ` ` `int` `a = ` `25` `, b = ` `35` `; ` ` ` `findSmallest(a, b); ` ` ` `} ` `} ` ` ` `// This code is contributed by Anant Agarwal. ` |

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## Python3

`# Python program to find the ` `# smallest values of x and y that ` `# satisfy "ax - by = 0" ` ` ` `# To find GCD using Eculcid's algorithm ` `def` `gcd(a, b): ` ` ` `if` `(b ` `=` `=` `0` `): ` ` ` `return` `a ` ` ` `return` `(gcd(b, a ` `%` `b)) ` ` ` `# Prints smallest values of x and y that ` `# satisfy "ax - by = 0" ` `def` `findSmallest(a, b): ` ` ` ` ` `# Find LCM ` ` ` `lcm ` `=` `(a ` `*` `b)` `/` `gcd(a, b) ` ` ` `print` `(` `"x ="` `, lcm ` `/` `a, ` `"\ny = "` `, lcm ` `/` `b) ` ` ` `# Driver code ` `a ` `=` `25` `b ` `=` `35` `findSmallest(a, b) ` ` ` `# This code is contributed ` `# by Anant Agarwal. ` |

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## C#

`// C# program to find the smallest ` `// values of x and y that ` `// satisfy "ax - by = 0" ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `// To find GCD using ` ` ` `// Eculcid's algorithm ` ` ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` ` ` `if` `(b == 0) ` ` ` `return` `a; ` ` ` `return` `(gcd(b, a % b)); ` ` ` `} ` ` ` ` ` `// Prints smallest values of x and ` ` ` `// y that satisfy "ax - by = 0" ` ` ` `static` `void` `findSmallest(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` ` ` `// Find LCM ` ` ` `int` `lcm = (a * b) / gcd(a, b); ` ` ` ` ` `Console.Write(` `"x = "` `+ lcm / a + ` `"\ny = "` `+ lcm / b); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` `int` `a = 25, b = 35; ` ` ` `findSmallest(a, b); ` ` ` `} ` `} ` ` ` `// This code is contributed by Sam007. ` |

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## PHP

`<?php ` `// PHP program to find the ` `// smallest values of x ` `// and y that satisfy ` `// "ax - by = 0" ` ` ` `// To find GCD using ` `// Eculcid's algorithm ` `function` `gcd(` `$a` `, ` `$b` `) ` `{ ` ` ` `if` `(` `$b` `== 0) ` ` ` `return` `$a` `; ` ` ` `return` `(gcd(` `$b` `, ` `$a` `% ` `$b` `)); ` `} ` ` ` `// Prints smallest values ` `// of x and y that ` `// satisfy "ax - by = 0" ` `function` `findSmallest(` `$a` `, ` `$b` `) ` `{ ` ` ` ` ` `// Find LCM ` ` ` `$lcm` `= (` `$a` `* ` `$b` `) / gcd(` `$a` `, ` `$b` `); ` ` ` ` ` `echo` `"x = "` `, ` `$lcm` `/` `$a` `, ` `"\ny = "` `, ` `$lcm` `/` `$b` `; ` `} ` ` ` ` ` `// Driver Code ` ` ` `$a` `= 25; ` ` ` `$b` `= 35; ` ` ` `findSmallest(` `$a` `, ` `$b` `); ` ` ` `// This code is contributed by ajit ` `?> ` |

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Output:

x = 7 y = 5

**The above code for findSmallest() can be reduced:**

Since ax - by = 0, ax = by, which means x/y = b/a So we can calculate gcd and directly do as - Value of x = b / gcd; Value of y = a / gcd;

`// Prints smallest values of x and y that ` `// satisfy "ax - by = 0" ` `void` `findSmallest(` `int` `a, ` `int` `b) ` `{ ` ` ` `// Find GCD ` ` ` `int` `g = gcd(a, b); ` ` ` ` ` `cout << ` `"x = "` `<< b / g ` ` ` `<< ` `"\ny = "` `<< a / g; ` `} ` |

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This article is contributed by **Aakash Sachdeva**. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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