# Check if a large number is divisible by 8 or not

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 = 1128Output : YesInput : n = 1124Output : NoInput  : n = 363588395960667043875487Output : 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 = 952Since 952 is divisible by 8, answer is YES.`

How does this work?

`Let us consider 76952, we can write it as76942 = 7*10000 + 6*1000 + 9*100 + 5*10 + 2The 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 = 52Therefore 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 ``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

 ` `

## PHP

 ` `

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 `` ` `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 ``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)