# Largest number by which given 3 numbers should be divided such that they leaves same remainder

Given three numbers, our task is to find the largest number by which when given 3 numbers are divided leads to same remainder. It may be assumed that all given numbers are given in increasing order.

Examples:

Input :a = 62, b = 132, c = 237Output :35 35 leads to same remainder 27 when divides 62, 132 and 237.Input :a = 74, b = 272, c = 584Output :6

The idea is based on the fact that if a number leaves same remainder with a, b and c, then it would divide their differences. Let us understand assuming that x is our result. Let **a = x*d1 + r** where r is the remainder when a is divided by x. Similarly we can write **b = x*d2 + r** and **b = x*d3 + r**. So the logic is here we first find differences of all three pairs and after that, we find greatest common divisor of differences to maximize result.

Below is the implementation of above idea.

## C++

`// C++ program to find the largest numbers that` `// leads to same remainder when divides given` `// three sorted numbers` `#include <bits/stdc++.h>` `using` `namespace` `std;` `//__gcd function` `int` `gcd(` `int` `a, ` `int` `b)` `{` ` ` `if` `(a == 0)` ` ` `return` `b;` ` ` `return` `gcd(b % a, a);` `}` `// function return number which divides these` `// three number and leaves same remainder .` `int` `sameRemainder(` `int` `a, ` `int` `b, ` `int` `c)` `{` ` ` `// We find the differences of all three pairs` ` ` `int` `a1 = (b - a), b1 = (c - b), c1 = (c - a);` ` ` `// Return GCD of three differences.` ` ` `return` `gcd(a1, gcd(b1, c1));` `}` `// driver program` `int` `main()` `{` ` ` `int` `a = 62, b = 132, c = 237;` ` ` `cout << sameRemainder(a, b, c) << endl;` ` ` `return` `0;` `}` |

## Java

`// Java program to find the largest` `// numbers that leads to same` `// remainder when divides given` `// three sorted numbers` `class` `GFG {` ` ` `//__gcd function` `static` `int` `gcd(` `int` `a, ` `int` `b)` `{` ` ` `if` `(a == ` `0` `)` ` ` `return` `b;` ` ` `return` `gcd(b % a, a);` `}` `// function return number which divides these` `// three number and leaves same remainder .` `static` `int` `sameRemainder(` `int` `a, ` `int` `b, ` `int` `c)` `{` ` ` `// We find the differences of all three pairs` ` ` `int` `a1 = (b - a), b1 = (c - b), c1 = (c - a);` ` ` `// Return GCD of three differences.` ` ` `return` `gcd(a1, gcd(b1, c1));` `}` `// Driver code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `a = ` `62` `, b = ` `132` `, c = ` `237` `;` ` ` `System.out.println(sameRemainder(a, b, c));` `}` `}` `// This code is contributed by Anant Agarwal.` |

## Python3

`# Python program to find` `# the largest numbers that` `# leads to same remainder` `# when divides given` `# three sorted numbers` `#__gcd function` `def` `gcd(a,b):` ` ` `if` `(a ` `=` `=` `0` `):` ` ` `return` `b` ` ` `return` `gcd(b ` `%` `a, a)` ` ` `# function return number` `# which divides these` `# three number and leaves` `# same remainder .` `def` `sameRemainder(a,b,c):` ` ` `# We find the differences` ` ` `# of all three pairs` ` ` `a1 ` `=` `(b ` `-` `a)` ` ` `b1 ` `=` `(c ` `-` `b)` ` ` `c1 ` `=` `(c ` `-` `a)` ` ` ` ` `# Return GCD of three differences.` ` ` `return` `gcd(a1, gcd(b1, c1))` `# Driver program` `a ` `=` `62` `b ` `=` `132` `c ` `=` `237` `print` `(sameRemainder(a, b, c))` `# This code is contributed` `# by Anant Agarwal.` |

## C#

`// C# program to find the largest` `// numbers that leads to same` `// remainder when divides given` `// three sorted numbers` `using` `System;` `class` `GFG {` ` ` `// gcd function` `static` `int` `gcd(` `int` `a, ` `int` `b)` `{` ` ` `if` `(a == 0)` ` ` `return` `b;` ` ` `return` `gcd(b % a, a);` `}` `// function return number which divides` `// these three number and leaves same` `// remainder .` `static` `int` `sameRemainder(` `int` `a, ` `int` `b, ` `int` `c)` `{` ` ` `// We find the differences of all three pairs` ` ` `int` `a1 = (b - a), b1 = (c - b), c1 = (c - a);` ` ` `// Return GCD of three differences.` ` ` `return` `gcd(a1, gcd(b1, c1));` `}` `// Driver code` `public` `static` `void` `Main()` `{` ` ` `int` `a = 62, b = 132, c = 237;` ` ` `Console.WriteLine(sameRemainder(a, b, c));` `}` `}` `// This code is contributed by vt_m.` |

## PHP

`<?php` `// PHP program to find the` `// largest numbers that leads` `// to same remainder when` `// divides given three sorted` `// numbers` `//__gcd function` `function` `gcd(` `$a` `, ` `$b` `)` `{` ` ` `if` `(` `$a` `== 0)` ` ` `return` `$b` `;` ` ` `return` `gcd(` `$b` `% ` `$a` `, ` `$a` `);` `}` `// function return number` `// which divides these` `// three number and` `// leaves same remainder .` `function` `sameRemainder( ` `$a` `, ` `$b` `, ` `$c` `)` `{` ` ` ` ` `// We find the differences` ` ` `// of all three pairs` ` ` `$a1` `= (` `$b` `- ` `$a` `);` ` ` `$b1` `= (` `$c` `- ` `$b` `);` ` ` `$c1` `= (` `$c` `- ` `$a` `);` ` ` `// Return GCD of` ` ` `// three differences.` ` ` `return` `gcd(` `$a1` `, gcd(` `$b1` `, ` `$c1` `));` `}` ` ` `// Driver Code` ` ` `$a` `= 62;` ` ` `$b` `= 132;` ` ` `$c` `= 237;` ` ` `echo` `sameRemainder(` `$a` `, ` `$b` `, ` `$c` `) ;` `// This code is contributed by anuj_67.` `?>` |

## Javascript

`<script>` `// Javascript program to find the largest` `// numbers that leads to same remainder` `// when divides given three sorted numbers` `//__gcd function` `function` `gcd(a, b)` `{` ` ` `if` `(a == 0)` ` ` `return` `b;` ` ` ` ` `return` `gcd(b % a, a);` `}` `// Function return number which divides these` `// three number and leaves same remainder .` `function` `sameRemainder(a, b, c)` `{` ` ` ` ` `// We find the differences of all three pairs` ` ` `var` `a1 = (b - a), b1 = (c - b), c1 = (c - a);` ` ` `// Return GCD of three differences.` ` ` `return` `gcd(a1, gcd(b1, c1));` `}` `// Driver code` `var` `a = 62, b = 132, c = 237;` `document.write(sameRemainder(a, b, c));` `// This code is contributed by noob2000` `</script>` |

**Output: **

35

**Time Complexity:** O( log N )

**Auxiliary Space:** O(1)