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Check if a large number is divisible by 8 or not

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Given a number, the task is to check if a number is divisible by 8 or not. The input number may be large and it may not be possible to store even if we use long long int.
Examples: 

Input  : n = 1128
Output : Yes
Input : n = 1124
Output : No
Input : n = 363588395960667043875487
Output : No

Since input number may be very large, we cannot use n % 8 to check if a number is divisible by 8 or not, especially in languages like C/C++. The idea is based on following fact.

A number is divisible by 8 if number formed by last three digits of it is divisible by 8.

Illustration: 

For example, let us consider 76952 
Number formed by last three digits = 952
Since 952 is divisible by 8, answer is YES.

How does this work? 

Let us consider 76952, we can write it as
76942 = 7*10000 + 6*1000 + 9*100 + 5*10 + 2
The proof is based on below observation:
Remainder of 10i divided by 8 is 0 if i greater
than or equal to three. Note than 10000,
1000,... etc lead to remainder 0 when divided by 8.
So remainder of "7*10000 + 6*1000 + 9*100 +
5*10 + 2" divided by 8 is equivalent to remainder
of following :
0 + 0 + 9*100 + 5*10 + 2 = 52
Therefore we can say that the whole number is
divisible by 8 if 952 is divisible by 8.


Below is the implementation of the above fact :

C++

// C++ program to find if a number is divisible by
// 8 or not
#include<bits/stdc++.h>
using namespace std;
  
// Function to find that number divisible by
// 8 or not
bool check(string str)
{
    int n = str.length();
  
    // Empty string
    if (n == 0)
        return false;
  
    // If there is single digit
    if (n == 1)
        return ((str[0]-'0')%8 == 0);
  
    // If there is double digit
    if (n == 2)
        return (((str[n-2]-'0')*10 + (str[n-1]-'0'))%8 == 0);
  
    // If number formed by last three digits is
    // divisible by 8.
    int last = str[n-1] - '0';
    int second_last = str[n-2] - '0';
    int third_last = str[n-3] - '0';
  
    return ((third_last*100 + second_last*10 + last) % 8 == 0);
}
  
// Driver code
int main()
{
    string str = "76952";
    check(str)?  cout << "Yes" : cout << "No ";
    return 0;
}

                    

Java

// Java program to find if a number is
// divisible by 8 or not
import java.io.*;
class IsDivisible
{
    // Function to find that number divisible by
    // 8 or not
    static boolean check(String str)
    {
        int n = str.length();
       
        // Empty string
        if (n == 0)
            return false;
       
        // If there is single digit
        if (n == 1)
            return ((str.charAt(0)-'0')%8 == 0);
       
        // If there is double digit
        if (n == 2)
            return (((str.charAt(n-2)-'0')*10 + (str.charAt(n-1)-'0'))%8 == 0);
       
        // If number formed by last three digits is
        // divisible by 8.
        int last = str.charAt(n-1) - '0';
        int second_last = str.charAt(n-2) - '0';
        int third_last = str.charAt(n-3) - '0';
       
        return ((third_last*100 + second_last*10 + last) % 8 == 0);
    }
      
    // main function
    public static void main (String[] args) 
    {
        String str = "76952";
        if(check(str))
            System.out.println("Yes");
        else
            System.out.println("No");
    }

                    

Python3

# Python 3 program to find 
# if a number is divisible
# by 8 or not
  
  
# Function to find that
# number divisible by 8
# or not
def check(st) :
    n = len(st)
      
    # Empty string
    if (n == 0) :
        return False
  
    # If there is single digit
    if (n == 1) :
        return ((int)(st[0]) % 8 == 0)
  
    # If there is double digit
    if (n == 2) :
        return ((int)(st[n - 2]) * 10 +
          ((int)(str[n - 1]) % 8 == 0))
  
    # If number formed by last
    # three digits is divisible
    # by 8.
    last = (int)(st[n - 1])
    second_last = (int)(st[n - 2])
    third_last = (int)(st[n - 3])
  
    return ((third_last*100 + second_last*10 +
                               last) % 8 == 0)
  
  
# Driver code
st = "76952"
  
if(check(st)) :
    print("Yes")
else
    print("No ")
      
# This code is contributed by Nikita tiwari

                    

C#

// C# program to find if a number
// is divisible by 8 or not
using System;
  
class IsDivisible
{
    // Function to find that number 
    // divisible by 8 or not
    static bool check(String str)
    {
        int n = str.Length;
      
        // Empty string
        if (n == 0)
            return false;
      
        // If there is single digit
        if (n == 1)
            return ((str[0] - '0') %8  == 0);
      
        // If there is double digit
        if (n == 2)
            return (((str[n - 2] - '0') * 10 + 
            (str[n - 1] - '0')) % 8 == 0);
      
        // If number formed by last three 
        // digits is divisible by 8
        int last = str[n - 1] - '0';
        int second_last = str[n - 2] - '0';
        int third_last = str[n - 3] - '0';
      
        return ((third_last * 100 + second_last  
                        * 10 + last) % 8 == 0);
    }
      
    // Driver Code
    public static void Main () 
    {
        String str = "76952";
        if(check(str))
            Console.Write("Yes");
        else
            Console.Write("No");
    }
  
// This Code is contributed by Nitin Mittal.

                    

Javascript

<script>
  
// JavaScript program for the above approach
  
    // Function to find that number  
    // divisible by 8 or not 
    function check(str) 
    
        let n = str.length; 
        
        // Empty string 
        if (n == 0) 
            return false
        
        // If there is single digit 
        if (n == 1) 
            return ((str[0] - '0') %8  == 0); 
        
        // If there is double digit 
        if (n == 2) 
            return (((str[n - 2] - '0') * 10 +  
            (str[n - 1] - '0')) % 8 == 0); 
        
        // If number formed by last three  
        // digits is divisible by 8 
        let last = str[n - 1] - '0'
        let second_last = str[n - 2] - '0'
        let third_last = str[n - 3] - '0'
        
        return ((third_last * 100 + second_last   
                        * 10 + last) % 8 == 0); 
    
  
// Driver Code
      
     let str = "76952"
     if(check(str)) 
        document.write("Yes"); 
     else
         document.write("No");
  
// This code is contributed by splevel62.
</script>

                    

PHP

<?php
// PHP program to find if a number 
// is divisible by 8 or not
  
// Function to find that number
// divisible by 8 or not
function check($str)
{
    $n = strlen($str);
  
    // Empty string
    if ($n == 0)
        return false;
  
    // If there is single digit
    if ($n == 1)
        return (($str[0] - '0') %
                          8 == 0);
  
    // If there is double digit
    if ($n == 2)
        return ((($str[$n - 2] - '0') * 
               10 + ($str[$n - 1] - '0')) 
                                % 8 == 0);
  
    // If number formed by last three 
    // digits is divisible by 8.
    $last = $str[$n - 1] - '0';
    $second_last = $str[$n - 2] - '0';
    $third_last = $str[$n - 3] - '0';
  
    return (($third_last * 100 + 
             $second_last * 10 + 
             $last) % 8 == 0);
}
  
// Driver code
$str = "76952";
$x = check($str)? "Yes" : "No ";
echo($x);
  
// This code is contributed by Ajit.
?>

                    

Output
Yes







Time Complexity: O(1), as we are not using any loops for traversing.
Auxiliary Space: O(1), as we are not using any extra space.

Another Approach(Using Bitwise Operator):

The binary representation of the number 8 is 1000. If a number is divisible by 8, it means that its last three bits in binary representation are 000. Therefore, we can check if a number is divisible by 8 by checking if its last three bits are 000 or not.

To do this, we perform a bitwise AND operation between the input number and 7 (which is 111 in binary and has its last three bits set to 0). If the result of the bitwise AND operation is 0, then the input number is divisible by 8, otherwise it is not divisible by 8.

Note that we use long long int data type to handle large numbers.

C++

#include <iostream>
  
bool isDivisibleBy8(long long int n) {
    return (n & 7) == 0; // 7 is 0111 in binary
}
  
int main() {
    long long int n = 76952;
  
    if (isDivisibleBy8(n)) {
        std::cout << n << " is divisible by 8\n";
    } else {
        std::cout << n << " is not divisible by 8\n";
    }
  
    return 0;
}

                    

Java

import java.util.*;
  
public class Main {
    // Function to check if a number is divisible by 8
    static boolean isDivisibleBy8(long n) {
        return (n & 7) == 0; // 7 is 0111 in binary
    }
  
    public static void main(String[] args) {
        long n = 76952;
  
        if (isDivisibleBy8(n)) {
            System.out.println(n + " is divisible by 8");
        } else {
            System.out.println(n + " is not divisible by 8");
        }
    }
}

                    

Python3

def isDivisibleBy8(n: int) -> bool:
    return (n & 7) == 0  # 7 is 0111 in binary
  
n = 76952
if isDivisibleBy8(n):
    print(n, "is divisible by 8")
else:
    print(n, "is not divisible by 8")

                    

C#

using System;
  
public class Program {
  // Function to check if a number is divisible by 8
  static bool isDivisibleBy8(long n) {
    return (n & 7) == 0; // 7 is 0111 in binary
  }
  
  public static void Main(string[] args) {
    long n = 76952;
  
    if (isDivisibleBy8(n)) {
      Console.WriteLine(n + " is divisible by 8");
    } else {
      Console.WriteLine(n + " is not divisible by 8");
    }
  }
}

                    

Javascript

function isDivisibleBy8(n) {
    return (n & 7) == 0; // 7 is 0111 in binary
}
  
let n = 76952;
if (isDivisibleBy8(n)) {
    console.log(n + " is divisible by 8");
} else {
    console.log(n + " is not divisible by 8");
}

                    

Output
76952 is divisible by 8







Time Complexity: O(1)
Auxiliary Space: O(1)

Another Approach (Using String) :

  1. We can reduce the number of checks by breaking the string into substrings of length 3.
  2. If a substring is divisible by 8, then the entire number is divisible by 8.
  3. We can check the divisibility of a substring of length 3 using the following formula:                                                                                   A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
  4. We can use modular arithmetic to compute the number formed by the last three digits of a substring.

C++

#include<bits/stdc++.h>
using namespace std;
  
// Function to find that number divisible by
// 8 or not
bool check(string str)
{
    int n = str.length();
  
    // Empty string
    if (n == 0)
        return false;
  
    // If there is single digit
    if (n == 1)
        return ((str[0]-'0')%8 == 0);
  
    // If there is double digit
    if (n == 2)
        return (((str[n-2]-'0')*10 + (str[n-1]-'0'))%8 == 0);
  
    // Check substrings of length 3
    for (int i = 0; i < n - 2; i++) {
        int num = (str[i]-'0')*100 + (str[i+1]-'0')*10 + (str[i+2]-'0');
        if (num % 8 == 0)
            return true;
    }
  
    // If number is not divisible by 8
    return false;
}
  
// Driver code
int main()
{
    string str = "76952";
    check(str)? cout << "Yes" : cout << "No";
    return 0;
}

                    

Java

import java.util.Scanner;
  
public class Main {
  
    // Function to find if a number is divisible by 8 or not
    static boolean check(String str) {
        int n = str.length();
  
        // Empty string
        if (n == 0)
            return false;
  
        // If there is a single digit
        if (n == 1)
            return ((str.charAt(0) - '0') % 8 == 0);
  
        // If there are two digits
        if (n == 2)
            return (((str.charAt(n - 2) - '0') * 10 + (str.charAt(n - 1) - '0')) % 8 == 0);
  
        // Check substrings of length 3
        for (int i = 0; i < n - 2; i++) {
            int num = (str.charAt(i) - '0') * 100 + (str.charAt(i + 1) - '0') * 10 + (str.charAt(i + 2) - '0');
            if (num % 8 == 0)
                return true;
        }
  
        // If number is not divisible by 8
        return false;
    }
  
    // Driver code
    public static void main(String[] args) {
        String str = "76952";
        if (check(str))
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}

                    

Python3

def check(str):
    n = len(str)
  
    # Empty string
    if n == 0:
        return False
  
    # If there is a single digit
    if n == 1:
        return int(str) % 8 == 0
  
    # If there are two digits
    if n == 2:
        return int(str) % 8 == 0
  
    # Check substrings of length 3
    for i in range(n - 2):
        num = int(str[i:i+3])
        if num % 8 == 0:
            return True
  
    # If number is not divisible by 8
    return False
  
# Driver code
str = "76952"
if check(str):
    print("Yes")
else:
    print("No")

                    

C#

using System;
  
public class MainClass
{
    // Function to find whether the number is divisible by 8 or not
    public static bool Check(string str)
    {
        int n = str.Length;
  
        // Empty string
        if (n == 0)
            return false;
  
        // If there is a single digit
        if (n == 1)
            return ((str[0] - '0') % 8 == 0);
  
        // If there are two digits
        if (n == 2)
            return (((str[n - 2] - '0') * 10 + (str[n - 1] - '0')) % 8 == 0);
  
        // Check substrings of length 3
        for (int i = 0; i < n - 2; i++)
        {
            int num = (str[i] - '0') * 100 + (str[i + 1] - '0') * 10 + (str[i + 2] - '0');
            if (num % 8 == 0)
                return true;
        }
  
        // If the number is not divisible by 8
        return false;
    }
  
    public static void Main(string[] args)
    {
        string str = "76952";
        Console.WriteLine(Check(str) ? "Yes" : "No");
    }
}

                    

Javascript

function check(str) {
  const n = str.length;
  
  // Empty string
  if (n === 0) {
    return false;
  }
  
  // If there is a single digit
  if (n === 1) {
    return parseInt(str) % 8 === 0;
  }
  
  // If there are two digits
  if (n === 2) {
    return parseInt(str) % 8 === 0;
  }
  
  // Check substrings of length 3
  for (let i = 0; i < n - 2; i++) {
    const num = parseInt(str.substr(i, 3));
    if (num % 8 === 0) {
      return true;
    }
  }
  
  // If the number is not divisible by 8
  return false;
}
  
const str = "76952";
console.log(check(str) ? "Yes" : "No");

                    

Output
Yes







In this code, we first check if the length of the string is 0, 1, or 2, and handle those cases separately. Then, we loop through the string and check substrings of length 3. If we find a substring that is divisible by 8, we return true. If we have checked all substrings and none of them is divisible by 8, we return false.

Time Complexity : O(n)

Space Complexity : O(1)


This article is contributed by DANISH_RAZA .

 



Last Updated : 06 Oct, 2023
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