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

Last Updated : 10 Apr, 2023

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 = 237
Output : 35
35 leads to same remainder 27 when divides
62, 132 and 237.

Input : a = 74, b = 272, c = 584
Output : 6```

Brute Force Approach:

The brute force approach to solve this problem is to start from the largest number among the three (i.e., c) and decrement it until we find a number that leaves the same remainder when divided by a and b.

We can find the remainder of a, b, and c when divided by any number n using the modulo operator (%). Then, we can check if the remainders of a, b, and c when divided by a candidate number n are equal. If they are equal, then n is a valid solution.

Below is the implementation of above approach:

## C++

 `// C++ program to find the largest numbers that` `// leads to same remainder when divides given` `// three sorted numbers` `#include ` `using` `namespace` `std;`   `// function return number which divides these` `// three number and leaves same remainder .` `int` `sameRemainder(``int` `a, ``int` `b, ``int` `c)` `{` `    ``// start from largest number and decrement until we find` `    ``// a solution` `    ``for` `(``int` `n = c; n >= 1; n--) {` `        ``int` `remainderA = a % n;` `        ``int` `remainderB = b % n;` `        ``int` `remainderC = c % n;` `        ``if` `(remainderA == remainderB` `            ``&& remainderB == remainderC) {` `            ``return` `n;` `        ``}` `    ``}` `    ``// no solution found` `    ``return` `-1;` `}`   `// 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` `import` `java.util.*;`   `public` `class` `Main {`   `    ``// function return number which divides these` `    ``// three number and leaves same remainder .` `    ``static` `int` `sameRemainder(``int` `a, ``int` `b, ``int` `c)` `    ``{` `        ``// start from largest number and decrement until we` `        ``// find a solution` `        ``for` `(``int` `n = c; n >= ``1``; n--) {` `            ``int` `remainderA = a % n;` `            ``int` `remainderB = b % n;` `            ``int` `remainderC = c % n;` `            ``if` `(remainderA == remainderB` `                ``&& remainderB == remainderC) {` `                ``return` `n;` `            ``}` `        ``}` `        ``// no solution found` `        ``return` `-``1``;` `    ``}`   `    ``// driver program` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``int` `a = ``62``, b = ``132``, c = ``237``;` `        ``System.out.println(sameRemainder(a, b, c));` `    ``}` `}`

## Python3

 `# Python program to find the largest numbers that` `# leads to same remainder when divides given` `# three sorted numbers`     `def` `same_remainder(a, b, c):` `    ``# start from largest number and decrement until we find a solution` `    ``for` `n ``in` `range``(c, ``0``, ``-``1``):` `        ``remainder_a ``=` `a ``%` `n` `        ``remainder_b ``=` `b ``%` `n` `        ``remainder_c ``=` `c ``%` `n` `        ``if` `remainder_a ``=``=` `remainder_b ``=``=` `remainder_c:` `            ``return` `n` `    ``# no solution found` `    ``return` `-``1`     `a ``=` `62` `b ``=` `132` `c ``=` `237` `print``(same_remainder(a, b, c))`

## C#

 `using` `System;`   `class` `Program ` `{`   `  ``// function return number which divides these` `  ``// three number and leaves same remainder .` `  ``static` `int` `SameRemainder(``int` `a, ``int` `b, ``int` `c)` `  ``{`   `    ``// start from largest number and decrement until we` `    ``// find a solution` `    ``for` `(``int` `n = c; n >= 1; n--) {` `      ``int` `remainderA = a % n;` `      ``int` `remainderB = b % n;` `      ``int` `remainderC = c % n;` `      ``if` `(remainderA == remainderB` `          ``&& remainderB == remainderC) {` `        ``return` `n;` `      ``}` `    ``}`   `    ``// no solution found` `    ``return` `-1;` `  ``}`   `  ``static` `void` `Main(``string``[] args)` `  ``{` `    ``int` `a = 62, b = 132, c = 237;` `    ``Console.WriteLine(SameRemainder(a, b, c));` `  ``}` `}`

## Javascript

 `function` `sameRemainder(a, b, c) {` `    ``// start from largest number and decrement until we find a solution` `    ``for` `(let n = c; n >= 1; n--) {` `        ``let remainderA = a % n;` `        ``let remainderB = b % n;` `        ``let remainderC = c % n;` `        ``if` `(remainderA == remainderB && remainderB == remainderC) {` `            ``return` `n;` `        ``}` `    ``}` `    ``// no solution found` `    ``return` `-1;` `}`   `// driver program` `let a = 62, b = 132, c = 237;` `console.log(sameRemainder(a, b, c));`

Output

`35`

Time Complexity: O(c)

Auxiliary Space: O(1)

Approach:

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 ` `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

 ``

## Javascript

 ``

Output

`35`

Time Complexity: O( log N )

Auxiliary Space: O(1)