Queries for GCD of all numbers of an array except elements in a given range

Given an array of n numbers and a number of queries are also given. Each query will be represented by two integers l, r. The task is to find out the GCD of all the numbers of the array excluding the numbers given in the range l, r (both inclusive) for each query.

Examples:

Input : arr[] = {2, 6, 9}
        Ranges [0 0], [1 1], [1 2]
Output : 3
         1
         2 
GCD of numbers excluding [0 0] or 
first element is GCD(6, 9) = 3
GCD of numbers excluding the [1 1] or
second element is GCD(2, 9) = 1
GCD of numbers excluding [1 2] is 
equal to first element = 2

Note : We use 1 based indexing in below explanation

We start from the very basic question how to calculate GCD of two numbers the best choice is Euclid’s algorithm . Now how to calculate GCD of more than one numbers, the solution is simple to suppose there are three numbers a, b and c. GCD(a, b, c) = GCD(GCD(a, b), c). In this way, we can easily find out GCD of any number of numbers.

One simple way to solve the question for each query suppose the range given is l and r. Take GCD of the numbers from 1 to l-1 suppose it is x then take GCD of the numbers from the range r+1 to n let it be y the output of each query will be GCD (x, y).
An efficient solution is to use two arrays, one as a prefix array and the second one as suffix array. The i-th index of the prefix array will store GCD of the numbers from 1 to i and the ith index of suffix array will denote the GCD of the numbers from i to n. Now suppose in a particular query range given is l, r it is obvious that the output for that query will be GCD(prefix[l-1], suffix[r+1]).

C++

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// C++ program for queries of GCD excluding
// given range of elements.
#include<bits/stdc++.h>
using namespace std;
  
// Calculating GCD using Euclid's algorithm
int GCD(int a, int b)
{
    if (b==0)
        return a;
    return GCD (b, a%b);
}
  
// Filling the prefix and suffix array
void FillPrefixSuffix(int prefix[], int arr[],
                           int suffix[], int n)
{
    // Filling the prefix array following relation
    // prefix(i) = GCD(prefix(i-1), arr(i))
    prefix[0] = arr[0];
    for (int i=1 ;i<n; i++)
        prefix[i] = GCD (prefix[i-1], arr[i]);
  
    // Filling the suffix array following the
    // relation suffix(i) = GCD(suffix(i+1), arr(i))
    suffix[n-1] = arr[n-1];
  
    for (int i=n-2; i>=0 ;i--)
        suffix[i] = GCD (suffix[i+1], arr[i]);
}
  
// To calculate gcd of the numbers outside range
int GCDoutsideRange(int l, int r, int prefix[],
                           int suffix[], int n)
{
    // If l=0, we need to tell GCD of numbers
    // from r+1 to n
    if (l==0)
        return suffix[r+1];
  
    // If r=n-1 we need to return the gcd of
    // numbers from 1 to l
    if (r==n-1)
        return prefix[l-1];
    return GCD(prefix[l-1], suffix[r+1]);
}
  
// Driver function
int main()
{
    int arr[] = {2, 6, 9};
    int n = sizeof(arr)/sizeof(arr[0]);
    int prefix[n], suffix[n];
    FillPrefixSuffix(prefix, arr, suffix, n);
  
    int l = 0, r = 0;
    cout << GCDoutsideRange(l, r, prefix, suffix, n)
         << endl;
    l = 1 ; r = 1;
    cout << GCDoutsideRange(l, r, prefix, suffix, n)
         << endl;
    l = 1 ; r = 2;
    cout << GCDoutsideRange(l, r, prefix, suffix, n)
         << endl;
    return 0;
}

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Java

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// Java program for queries of GCD excluding
// given range of elements.
import java.util.*;
  
class GFG {
      
// Calculating GCD using euclid algorithm
static int GCD(int a, int b)
{
    if (b == 0)
    return a;
    return GCD(b, a % b);
}
  
// Filling the prefix and suffix array
static void FillPrefixSuffix(int prefix[], 
          int arr[], int suffix[], int n) 
{
    // Filling the prefix array following relation
    // prefix(i) = GCD(prefix(i-1), arr(i))
    prefix[0] = arr[0];
    for (int i = 1; i < n; i++)
    prefix[i] = GCD(prefix[i - 1], arr[i]);
  
    // Filling the suffix array folowing the
    // relation suffix(i) = GCD(suffix(i+1), arr(i))
    suffix[n - 1] = arr[n - 1];
  
    for (int i = n - 2; i >= 0; i--)
    suffix[i] = GCD(suffix[i + 1], arr[i]);
}
  
// To calculate gcd of the numbers outside range
static int GCDoutsideRange(int l, int r,
      int prefix[], int suffix[], int n) {
            
    // If l=0, we need to tell GCD of numbers
    // from r+1 to n
    if (l == 0)
    return suffix[r + 1];
  
    // If r=n-1 we need to return the gcd of
    // numbers from 1 to l
    if (r == n - 1)
    return prefix[l - 1];
    return GCD(prefix[l - 1], suffix[r + 1]);
}
  
// Driver code
public static void main(String[] args) {
    int arr[] = {2, 6, 9};
    int n = arr.length;
    int prefix[] = new int[n];
    int suffix[] = new int[n];
    FillPrefixSuffix(prefix, arr, suffix, n);
  
    int l = 0, r = 0;
    System.out.println(GCDoutsideRange
             (l, r, prefix, suffix, n));
  
    l = 1;
    r = 1;
    System.out.println(GCDoutsideRange
             (l, r, prefix, suffix, n));
  
    l = 1;
    r = 2;
    System.out.println(GCDoutsideRange
             (l, r, prefix, suffix, n));
}
}
  
// This code is contributed by Anant Agarwal.

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Python3

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# Python program for
# queries of GCD excluding
# given range of elements.
  
# Calculating GCD
# using euclid algorithm
def GCD(a,b):
    if (b==0):
        return a
    return GCD (b, a%b)
  
   
# Filling the prefix
# and suffix array
def FillPrefixSuffix(prefix,arr,suffix,n):
  
    # Filling the prefix array
    # following relation
    # prefix(i) = GCD(prefix(i-1), arr(i))
    prefix[0] = arr[0]
    for i in range(1,n):
        prefix[i] = GCD (prefix[i-1], arr[i])
   
    # Filling the suffix
    # array folowing the
    # relation suffix(i) = GCD(suffix(i+1), arr(i))
    suffix[n-1] = arr[n-1]
   
    for i in range(n-2,-1,-1):
        suffix[i] = GCD (suffix[i+1], arr[i])
  
   
# To calculate gcd of
# the numbers outside range
def GCDoutsideRange(l,r,prefix,suffix,n):
      
    # If l=0, we need to tell GCD of numbers
    # from r+1 to n
    if (l==0):
        return suffix[r+1]
   
    # If r=n-1 we need to return the gcd of
    # numbers from 1 to l
    if (r==n-1):
        return prefix[l-1]
    return GCD(prefix[l-1], suffix[r+1])
  
# Driver code
  
arr = [2, 6, 9]
n = len(arr)
prefix=[]
suffix=[]
for i in range(n+1):
    prefix.append(0)
    suffix.append(0)
      
FillPrefixSuffix(prefix, arr, suffix, n)
l = 0 
r = 0
print(GCDoutsideRange(l, r, prefix, suffix, n))
  
l = 1
r = 1
print(GCDoutsideRange(l, r, prefix, suffix, n))
  
l = 1
r = 2
print(GCDoutsideRange(l, r, prefix, suffix, n))
  
# This code is contributed
# by Anant Agarwal.

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C#

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// C# program for queries of GCD 
// excluding given range of elements.
using System;
  
class GFG {
      
// Calculating GCD using 
// euclid algorithm
static int GCD(int a, int b)
{
    if (b == 0)
    return a;
    return GCD(b, a % b);
}
  
// Filling the prefix and suffix array
static void FillPrefixSuffix(int []prefix, 
                             int []arr, 
                             int []suffix,
                             int n) 
{
      
    // Filling the prefix array following
    // relation prefix(i) = 
    // GCD(prefix(i - 1), arr(i))
    prefix[0] = arr[0];
    for (int i = 1; i < n; i++)
    prefix[i] = GCD(prefix[i - 1], arr[i]);
  
    // Filling the suffix array folowing 
    // the relation suffix(i) = 
    // GCD(suffix(i+1), arr(i))
    suffix[n - 1] = arr[n - 1];
  
    for (int i = n - 2; i >= 0; i--)
    suffix[i] = GCD(suffix[i + 1], arr[i]);
}
  
// To calculate gcd of the numbers outside range
static int GCDoutsideRange(int l, int r,
                           int []prefix, 
                           int []suffix,
                           int n)
    {
              
    // If l=0, we need to tell 
    // GCD of numbers from r+1 to n
    if (l == 0)
    return suffix[r + 1];
  
    // If r=n-1 we need to return the 
    // gcd of numbers from 1 to l
    if (r == n - 1)
    return prefix[l - 1];
    return GCD(prefix[l - 1], suffix[r + 1]);
}
  
// Driver Code
public static void Main()
{
    int []arr = {2, 6, 9};
    int n = arr.Length;
    int []prefix = new int[n];
    int []suffix = new int[n];
    FillPrefixSuffix(prefix, arr, suffix, n);
  
    int l = 0, r = 0;
    Console.WriteLine(GCDoutsideRange
                     (l, r, prefix, suffix, n));
   
    l = 1;
    r = 1;
    Console.WriteLine(GCDoutsideRange
                        (l, r, prefix, suffix, n));
  
    l = 1;
    r = 2;
    Console.Write(GCDoutsideRange
                 (l, r, prefix, suffix, n));
}
}
  
// This code is contributed by Nitin Mittal.

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PHP

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<?php 
// PHP program for queries of GCD excluding
// given range of elements.
  
// Calculating GCD using euclid algorithm
function GCD($a, $b)
{
    if ($b == 0)
        return $a;
    return GCD ($b, $a % $b);
}
  
// Filling the prefix and suffix array
function FillPrefixSuffix(&$prefix, &$arr, &$suffix, $n)
{
    // Filling the prefix array following relation
    // prefix(i) = GCD(prefix(i-1), arr(i))
    $prefix[0] = $arr[0];
    for ($i = 1; $i < $n; $i++)
        $prefix[$i] = GCD ($prefix[$i - 1], $arr[$i]);
  
    // Filling the suffix array folowing the
    // relation suffix(i) = GCD(suffix(i+1), arr(i))
    $suffix[$n - 1] = $arr[$n - 1];
  
    for ($i = $n - 2; $i >= 0 ;$i--)
        $suffix[$i] = GCD ($suffix[$i + 1], $arr[$i]);
}
  
// To calculate gcd of the numbers outside range
function GCDoutsideRange($l, $r, &$prefix, &$suffix, $n)
{
    // If l=0, we need to tell GCD of numbers
    // from r+1 to n
    if ($l == 0)
        return $suffix[$r + 1];
  
    // If r=n-1 we need to return the gcd of
    // numbers from 1 to l
    if ($r == $n - 1)
        return $prefix[$l - 1];
    return GCD($prefix[$l - 1], $suffix[$r + 1]);
}
  
// Driver Code
$arr = array(2, 6, 9);
$n = sizeof($arr);
$prefix = array_fill(0, $n, NULL);
$suffix = array_fill(0, $n, NULL);
FillPrefixSuffix($prefix, $arr, $suffix, $n);
  
$l = 0;
$r = 0;
echo GCDoutsideRange($l, $r, $prefix, $suffix, $n) . "\n";
$l = 1 ;
$r = 1;
echo GCDoutsideRange($l, $r, $prefix, $suffix, $n) . "\n";
$l = 1 ;
$r = 2;
echo GCDoutsideRange($l, $r, $prefix, $suffix, $n) . "\n";
  
// This code is contributed by ita_c
?>

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Output:

3
1
2

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