# Check if a number is divisible by 23 or not

Given a number, the task is to quickly check if the number is divisible by 23 or not.**Examples:**

Input : x = 46 Output : Yes Input : 47 Output : No

A solution to the problem is to extract the last digit and add 7 times of last digit to remaining number and repeat this process until a two digit number is obtained. If the obtained two digit number is divisible by 23, then the given number is divisible by 23.**Approach:**

- Extract the last digit of the number/truncated number every time
- Add 7*(last digit of the previous number) to the truncated number
- Repeat the above three steps as long as necessary.

Illustration:

17043-->1704+7*3 = 1725-->172+7*5 = 207 which is 9*23, so 17043 is also divisible by 23.

Mathematical Proof :

Let be any number such that =100a+10b+c .

Now assume that is divisible by 23. Then

0 (mod 23)

100a+10b+c0 (mod 23)

10(10a+b)+c0 (mod 23)

10+c0 (mod 23)

Now that we have separated the last digit from the number, we have to find a way to use it.

Make the coefficient of 1.

In other words, we have to find an integer such that n such that 10n1 mod 23.

It can be observed that the smallest n which satisfies this property is 7 as 701 mod 23.

Now we can multiply the original equation 10+c0 (mod 23)

by 7 and simplify it:

70+7c0 (mod 23)

+7c0 (mod 23)

We have found out that if 0 (mod 23) then,

+7c0 (mod 23).

In other words, to check if a 3-digit number is divisible by 23,

we can just remove the last digit, multiply it by 7,

and then subtract it from the rest of the two digits.

## C++

`// CPP program to validate above logic` `#include <iostream>` `using` `namespace` `std;` `// Function to check if the number is` `// divisible by 23 or not` `bool` `isDivisible(` `long` `long` `int` `n)` `{` ` ` `// While there are at least 3 digits` ` ` `while` `(n / 100)` ` ` `{` ` ` `int` `d = n % 10; ` `// Extracting the last digit` ` ` `n /= 10; ` `// Truncating the number` ` ` `// Adding seven times the last` ` ` `// digit to the remaining number` ` ` `n += d * 7;` ` ` `}` ` ` `return` `(n % 23 == 0);` `}` `int` `main()` `{` ` ` `long` `long` `int` `n = 1191216;` ` ` `if` `(isDivisible(n))` ` ` `cout << ` `"Yes"` `<< endl;` ` ` `else` ` ` `cout << ` `"No"` `<< endl;` ` ` `return` `0;` `}` |

## Java

`// Java program to validate above logic` `class` `GFG` `{` ` ` `// Function to check if the` `// number is divisible by` `// 23 or not` `static` `boolean` `isDivisible(` `long` `n)` `{` ` ` `// While there are at` ` ` `// least 3 digits` ` ` `while` `(n / ` `100` `!= ` `0` `)` ` ` `{` ` ` `// Extracting the last digit` ` ` `long` `d = n % ` `10` `;` ` ` ` ` `n /= ` `10` `; ` `// Truncating the number` ` ` `// Adding seven times the last` ` ` `// digit to the remaining number` ` ` `n += d * ` `7` `;` ` ` `}` ` ` `return` `(n % ` `23` `== ` `0` `);` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `long` `n = ` `1191216` `;` ` ` `if` `(isDivisible(n))` ` ` `System.out.println(` `"Yes"` `);` ` ` `else` ` ` `System.out.println(` `"No"` `);` `}` `}` `// This code is contributed by mits` |

## Python 3

`# Python 3 program to validate above logic` `# Function to check if the number is` `# divisible by 23 or not` `def` `isDivisible(n) :` ` ` `# While there are at least 3 digits` ` ` `while` `n ` `/` `/` `100` `:` ` ` `# Extracting the last` ` ` `d ` `=` `n ` `%` `10` ` ` `# Truncating the number` ` ` `n ` `/` `/` `=` `10` ` ` `# Adding seven times the last ` ` ` `# digit to the remaining number` ` ` `n ` `+` `=` `d ` `*` `7` ` ` `return` `(n ` `%` `23` `=` `=` `0` `)` `# Driver Code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `n ` `=` `1191216` ` ` `# function calling` ` ` `if` `(isDivisible(n)) :` ` ` `print` `(` `"Yes"` `)` ` ` `else` `:` ` ` `print` `(` `"No"` `)` `# This code is contributed by ANKITRAI1` |

## C#

`// C# program to validate` `// above logic` `class` `GFG` `{` ` ` `// Function to check if the` `// number is divisible by` `// 23 or not` `static` `bool` `isDivisible(` `long` `n)` `{` ` ` `// While there are at` ` ` `// least 3 digits` ` ` `while` `(n / 100 != 0)` ` ` `{` ` ` `// Extracting the last digit` ` ` `long` `d = n % 10;` ` ` ` ` `n /= 10; ` `// Truncating the number` ` ` `// Adding seven times the last` ` ` `// digit to the remaining number` ` ` `n += d * 7;` ` ` `}` ` ` `return` `(n % 23 == 0);` `}` `// Driver Code` `public` `static` `void` `Main()` `{` ` ` `long` `n = 1191216;` ` ` `if` `(isDivisible(n))` ` ` `System.Console.WriteLine(` `"Yes"` `);` ` ` `else` ` ` `System.Console.WriteLine(` `"No"` `);` `}` `}` `// This code is contributed by mits` |

## PHP

`<?php` `// PHP program to validate above logic` `// Function to check if the number` `// is divisible by 23 or not` `function` `isDivisible(` `$n` `)` `{` ` ` `// While there are at` ` ` `// least 3 digits` ` ` `while` `(` `intval` `(` `$n` `/ 100))` ` ` `{` ` ` `$n` `= ` `intval` `(` `$n` `);` ` ` `$d` `= ` `$n` `% 10; ` `// Extracting the last digit` ` ` `$n` `/= 10; ` `// Truncating the number` ` ` `// Adding seven times the last` ` ` `// digit to the remaining number` ` ` `$n` `+= ` `$d` `* 7;` ` ` `}` ` ` `return` `(` `$n` `% 23 == 0);` `}` `$n` `= 1191216;` `if` `(isDivisible(` `$n` `))` `echo` `"Yes"` `. ` `"\n"` `;` `else` `echo` `"No"` `. ` `"\n"` `;` `// This code is contributed` `// by ChitraNayal` `?>` |

## Javascript

`<script>` `// JavaScript program to validate above logic` `// Function to check if the` `// number is divisible by` `// 23 or not` `function` `isDivisible(n)` `{` ` ` `// While there are at` ` ` `// least 3 digits` ` ` `while` `(Math.floor(n / 100) != 0)` ` ` `{` ` ` `// Extracting the last digit` ` ` `let d = n % 10;` ` ` ` ` `n = Math.floor(n/10); ` `// Truncating the number` ` ` ` ` `// Adding seven times the last` ` ` `// digit to the remaining number` ` ` `n += d * 7;` ` ` `}` ` ` ` ` `return` `(n % 23 == 0);` `}` `// Driver Code` `let n = 1191216;` `if` `(isDivisible(n))` ` ` `document.write(` `"Yes"` `);` `else` ` ` `document.write(` `"No"` `);` ` ` `// This code is contributed by rag2127` `</script>` |

**Output:**

Yes

Note that the above program may not make a lot of sense as could simply do n % 23 to check for divisibility. The idea of this program is to validate the concept. Also, this might be an efficient approach if input number is large and given as string.

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