Encontrar ‘k’ tal que su módulo con cada elemento de la array sea el mismo

Dada una array de n enteros, necesitamos encontrar todas las ‘k’ tales que 

arr[0] % k = arr[1] % k = ....... = arr[n-1] % k 

Ejemplos: 

Input  : arr[] = {6, 38, 34}
Output : 1 2 4
        6%1 = 38%1 = 34%1 = 0
        6%2 = 38%2 = 34%2 = 0
        6%4 = 38%4 = 34%2 = 2

Input  : arr[] = {3, 2}
Output : 1

Suponga que la array contiene solo dos elementos a y b (b>a). Entonces podemos escribir b = a + d donde d es un entero positivo y ‘k’ es un número tal que b%k = a%k. 

(a + d)%k = a%k
a%k + d%k = a%k 
d%k = 0

Ahora, lo que obtenemos del cálculo anterior es que ‘k’ debe ser un divisor de la diferencia entre los dos números. 

Ahora lo que tenemos que hacer cuando tenemos una array de enteros 

1. Averigüe la diferencia ‘d’ entre el elemento máximo y mínimo de la array
2. Descubre todos los divisores de ‘d’
3. Paso 3: para cada divisor, compruebe si arr[i]%divisor(d) es igual o no. Si es igual, imprímalo.

Implementación:

// C++ implementation of finding all k
// such that arr[i]%k is same for each i
#include<bits/stdc++.h>
using namespace std;
 
// Prints all k such that arr[i]%k is same for all i
void printEqualModNumbers (int arr[], int n)
{
    // sort the numbers
    sort(arr, arr + n);
 
    // max difference will be the difference between
    // first and last element of sorted array
    int d = arr[n-1] - arr[0];
     
    // Case when all the array elements are same
    if(d==0){
        cout<<"Infinite solution";
        return;
    }
 
    // Find all divisors of d and store in
    // a vector v[]
    vector <int> v;
    for (int i=1; i*i<=d; i++)
    {
        if (d%i == 0)
        {
            v.push_back(i);
            if (i != d/i)
                v.push_back(d/i);
        }
    }
 
    // check for each v[i] if its modulus with
    // each array element is same or not
    for (int i=0; i<v.size(); i++)
    {
        int temp = arr[0]%v[i];
 
        // checking for each array element if
        // its modulus with k is equal to k or not
        int j;
        for (j=1; j<n; j++)
            if (arr[j] % v[i] != temp)
                break;
 
        // if check is true print v[i]
        if (j == n)
            cout << v[i] <<" ";
    }
}
 
// Driver function
int main()
{
    int arr[] = {38, 6, 34};
    int n = sizeof(arr)/sizeof(arr[0]);
    printEqualModNumbers(arr, n);
    return 0;
}

//  Java implementation of finding all k
// such that arr[i]%k is same for each i
 
import java.util.Arrays;
import java.util.Vector;
 
class Test
{
    // Prints all k such that arr[i]%k is same for all i
    static void printEqualModNumbers (int arr[], int n)
    {
        // sort the numbers
        Arrays.sort(arr);
      
        // max difference will be the difference between
        // first and last element of sorted array
        int d = arr[n-1] - arr[0];
        // Case when all the array elements are same
        if(d==0){
            System.out.println("Infinite solution");
            return;
        }
        // Find all divisors of d and store in
        // a vector v[]
        Vector<Integer> v = new Vector<>();
        for (int i=1; i*i<=d; i++)
        {
            if (d%i == 0)
            {
                v.add(i);
                if (i != d/i)
                    v.add(d/i);
            }
        }
      
        // check for each v[i] if its modulus with
        // each array element is same or not
        for (int i=0; i<v.size(); i++)
        {
            int temp = arr[0]%v.get(i);
      
            // checking for each array element if
            // its modulus with k is equal to k or not
            int j;
            for (j=1; j<n; j++)
                if (arr[j] % v.get(i) != temp)
                    break;
      
            // if check is true print v[i]
            if (j == n)
                System.out.print(v.get(i) + " ");
        }
    }
     
    // Driver method
    public static void main(String args[])
    {
        int arr[] = {38, 6, 34};
         
        printEqualModNumbers(arr, arr.length);
    }
}

# Python3 implementation of finding all k
# such that arr[i]%k is same for each i
 
# Prints all k such that arr[i]%k is
# same for all i
def printEqualModNumbers(arr, n):
     
    # sort the numbers
    arr.sort();
     
    # max difference will be the difference
    # between first and last element of
    # sorted array
    d = arr[n - 1] - arr[0];
    // Case when all the array elements are same
    if(d==0):
        print("Infinite solution")
        return
     
    # Find all divisors of d and store
    # in a vector v[]
    v = [];
    i = 1;
    while (i * i <= d):
        if (d % i == 0):
                v.append(i);
                if (i != d / i):
                    v.append(d / i);
        i += 1;
     
    # check for each v[i] if its modulus with
    # each array element is same or not
    for i in range(len(v)):
        temp = arr[0] % v[i];
     
        # checking for each array element if
        # its modulus with k is equal to k or not
        j = 1;
        while (j < n):
            if (arr[j] % v[i] != temp):
                break;
            j += 1;
 
        # if check is true print v[i]
        if (j == n):
            print(v[i], end = " ");
 
# Driver Code
arr = [38, 6, 34];
printEqualModNumbers(arr, len(arr));
         
# This code is contributed by mits

// C# implementation of finding all k
// such that arr[i]%k is same for each i
using System;
using System.Collections;
class Test
{
    // Prints all k such that arr[i]%k is same for all i
    static void printEqualModNumbers (int []arr, int n)
    {
        // sort the numbers
        Array.Sort(arr);
     
        // max difference will be the difference between
        // first and last element of sorted array
        int d = arr[n-1] - arr[0];
        // Case when all the array elements are same
        if(d==0){
            Console.write("Infinite solution");
            return;
        }
        // Find all divisors of d and store in
        // a vector v[]
        ArrayList v = new ArrayList();
        for (int i=1; i*i<=d; i++)
        {
            if (d%i == 0)
            {
                v.Add(i);
                if (i != d/i)
                    v.Add(d/i);
            }
        }
     
        // check for each v[i] if its modulus with
        // each array element is same or not
        for (int i=0; i<v.Count; i++)
        {
            int temp = arr[0]%(int)v[i];
     
            // checking for each array element if
            // its modulus with k is equal to k or not
            int j;
            for (j=1; j<n; j++)
                if (arr[j] % (int)v[i] != temp)
                    break;
     
            // if check is true print v[i]
            if (j == n)
                Console.Write(v[i] + " ");
        }
    }
     
    // Driver method
    public static void Main()
    {
        int []arr = {38, 6, 34};
         
        printEqualModNumbers(arr, arr.Length);
    }
}
// This code is contributed by mits

<?php
// PHP implementation of finding all k
// such that arr[i]%k is same for each i
 
    // Prints all k such that arr[i]%k is same for all i
    function printEqualModNumbers ($arr, $n)
    {
        // sort the numbers
        sort($arr);
     
        // max difference will be the difference between
        // first and last element of sorted array
        $d = $arr[$n-1] - $arr[0];
        // Case when all the array elements are same
        if(d==0){
            print("Infinite solution");
            return;
        }
        // Find all divisors of d and store in
        // a vector v[]
        $v = array();
        for ($i=1; $i*$i<=$d; $i++)
        {
            if ($d%$i == 0)
            {
                array_push($v,$i);
                if ($i != $d/$i)
                    array_push($v,$d/$i);
            }
        }
     
        // check for each v[i] if its modulus with
        // each array element is same or not
        for ($i=0; $i<count($v); $i++)
        {
            $temp = $arr[0]%$v[$i];
     
            // checking for each array element if
            // its modulus with k is equal to k or not
            $j=1;
            for (; $j<$n; $j++)
                if ($arr[$j] % $v[$i] != $temp)
                    break;
     
            // if check is true print v[i]
            if ($j == $n)
                print($v[$i]." ");
        }
    }
     
    // Driver method
     
        $arr = array(38, 6, 34);
         
        printEqualModNumbers($arr, count($arr));
         
// This code is contributed by mits
?>

<script>
 
// JavaScript implementation of finding all k
// such that arr[i]%k is same for each i
 
    // Prints all k such that arr[i]%k is same for all i
    function printEqualModNumbers (arr, n)
    {
        // sort the numbers
        arr.sort((a, b) => a - b);
     
        // max difference will be the difference between
        // first and last element of sorted array
        d = arr[n-1] - arr[0];
        // Case when all the array elements are same
        if(d==0){
            document.write("Infinite solution");
            return;
        }
        // Find all divisors of d and store in
        // a vector v[]
        v = new Array();
        for (i=1; i*i<=d; i++)
        {
            if (d%i == 0)
            {
                v.push(i);
                if (i != d/i)
                    v.push(d/i);
            }
        }
     
        // check for each v[i] if its modulus with
        // each array element is same or not
        for (i=0; i< v.length; i++)
        {
            temp = arr[0]%v[i];
     
            // checking for each array element if
            // its modulus with k is equal to k or not
            j=1;
            for (; j<n; j++)
                if (arr[j] % v[i] != temp)
                    break;
     
            // if check is true print v[i]
            if (j == n)
                document.write(v[i] + " ");
        }
    }
     
    // Driver method
     
        let arr = new Array(38, 6, 34);
         
        printEqualModNumbers(arr, arr.length);
         
// This code is contributed by _saurabh_jaiswal
 
</script>

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