# 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;` `}` |

## 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.` |

## 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.` |

## 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.` |

## 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` `?>` |

## Javascript

`<script>` ` ` `// Javascript 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` ` ` `let lcm = parseInt((a * b) / gcd(a, b), 10);` ` ` ` ` `document.write(` `"x = "` `+ parseInt(lcm / a, 10) +` ` ` `"</br>y = "` `+ parseInt(lcm / b, 10));` ` ` `}` ` ` ` ` `let a = 25, b = 35;` ` ` `findSmallest(a, b);` ` ` `</script>` |

**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;

## C

`// 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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