Check if any large number is divisible by 17 or not

Difficulty Level : Easy
Last Updated : 27 Aug, 2021

Given a number, the task is to quickly check if the number is divisible by 17 or not.
Example:

Input : x = 34
Output : Yes

Input : x = 47
Output : No

A solution to the problem is to extract the last digit and subtract 5 times of the last digit from the remaining number and repeat this process until a two-digit number is obtained. If the obtained two-digit number is divisible by 17, then the given number is divisible by 17.
Approach:

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Extract the last digit of the number/truncated number every time
Subtract 5*(last digit of the previous number) from the truncated number
Repeat the above three steps as long as necessary.

Illustration:

3978-->397-5*8=357-->35-5*7=0. 
So 3978 is divisible by 17.

Mathematical Proof :
Let $\overline{a b c}$ be any number such that $\overline{a b c}$ =100a+10b+c .
Now assume that $\overline{a b c}$ is divisible by 17. Then
$\overline{a b c}\equiv$ 0 (mod 17)
100a+10b+c $\equiv$ 0 (mod 17)
10(10a+b)+c $\equiv$ 0 (mod 17)
10 $\overline{a b}$ +c $\equiv$ 0 (mod 17)
Now that we have separated the last digit from the number, we have to find a way to use it.
Make the coefficient of $\overline{a b}$ 1.
In other words, we have to find an integer such that n such that 10n $\equiv$ 1 mod 17.
It can be observed that the smallest n which satisfies this property is -5 as -50 $\equiv$ 1 mod 17.
Now we can multiply the original equation 10 $\overline{a b}$ +c $\equiv$ 0 (mod 17)
by -5 and simplify it:
-50 $\overline{a b}$ -5c $\equiv$ 0 (mod 17)
$\overline{a b}$ -5c $\equiv$ 0 (mod 17)
We have found out that if $\overline{a b c}\equiv$ 0 (mod 17) then,
$\overline{a b}$ -5c $\equiv$ 0 (mod 17).
In other words, to check if a 3-digit number is divisible by 17,
we can just remove the last digit, multiply it by 5,
and then subtract it from the rest of the two digits.

Program :

C++

// CPP Program to validate the above logic
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to check if the
// number is divisible by 17 or not
bool isDivisible(long long int n)
{
 
    while (n / 100)
    {
        // Extracting the last digit
        int d = n % 10;
 
        // Truncating the number
        n /= 10;
 
        // Subtracting the five times the
        // last digit from the remaining number
        n -= d * 5;
    }
 
    // Return n is divisible by 17
    return (n % 17 == 0);
}
 
// Driver code
int main()
{
    long long int n = 19877658;
    if (isDivisible(n))
        cout << "Yes" << endl;
    else
        cout << "No" << endl;
    return 0;
}

Java

// Java Program to validate the above logic
 
import java.io.*;
 
class GFG {
 
 
// Function to check if the
// number is divisible by 17 or not
 static boolean isDivisible(long n)
{
 
    while (n / 100>0)
    {
        // Extracting the last digit
        long d = n % 10;
 
        // Truncating the number
        n /= 10;
 
        // Subtracting the five times the
        // last digit from the remaining number
        n -= d * 5;
    }
 
    // Return n is divisible by 17
    return (n % 17 == 0);
}
 
// Driver code
 
    public static void main (String[] args) {
    long n = 19877658;
    if (isDivisible(n))
        System.out.println( "Yes");
    else
        System.out.println( "No");
    }
}
// This code is contributed by inder_verma.

Python 3

# Python 3 Program to validate
# the above logic
 
# Function to check if the
# number is divisible by 17 or not
def isDivisible(n) :
 
    while(n // 100) :
 
        # Extracting the last digit
        d = n % 10
 
        # Truncating the number
        n //= 10
 
        # Subtracting the five times 
        # the last digit from the
        # remaining number
        n -= d * 5
 
    # Return n is divisible by 17
    return (n % 17 == 0)
 
# Driver Code
if __name__ == "__main__" :
 
    n = 19877658
     
    if isDivisible(n) :
        print("Yes")
    else :
        print("No")
 
# This code is contributed
# by ANKITRAI1

C#

// C# Program to validate the above logic
  
using System;
  
class GFG {
  
  
// Function to check if the
// number is divisible by 17 or not
 static bool isDivisible(long n)
{
  
    while (n / 100>0)
    {
        // Extracting the last digit
        long d = n % 10;
  
        // Truncating the number
        n /= 10;
  
        // Subtracting the five times the
        // last digit from the remaining number
        n -= d * 5;
    }
  
    // Return n is divisible by 17
    return (n % 17 == 0);
}
  
// Driver code
  
    public static void Main () {
    long n = 19877658;
    if (isDivisible(n))
        Console.Write( "Yes");
    else
        Console.Write( "No");
    }
}

PHP

<?php
// PHP Program to validate the above logic
 
// Function to check if the
// number is divisible by 17 or not
function isDivisible($n)
{
 
    while ($n / 100 != 0)
    {
        // Extracting the last digit
        $d = (int)$n % 10;
 
        // Truncating the number
        $n /= 10;
 
        // Subtracting the five times
        // the last digit from the
        // remaining number
        $n -= $d * 5;
    }
 
    // Return n is divisible by 17
    return ($n % 17 == 0);
}
 
// Driver code
$n = 19877658;
if (isDivisible($n))
    print("Yes");
else
    print("No");
 
// This code is contributed by Raj
?>

Javascript

<script>
 
// JavaScript Program to validate the above logic
     
    // Function to check if the
// number is divisible by 17 or not
    function isDivisible(n)
    {
        while (Math.floor(n / 100)>0)
    {
        // Extracting the last digit
        let d = n % 10;
   
        // Truncating the number
        n = Math.floor(n/10);
   
        // Subtracting the five times the
        // last digit from the remaining number
        n -= d * 5;
    }
   
    // Return n is divisible by 17
    return (n % 17 == 0);
    }
     
    // Driver code
    let n = 19877658;
    if (isDivisible(n))
        document.write( "Yes");
    else
        document.write( "No");
         
 
 
// This code is contributed by avanitrachhadiya2155
 
</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.

My Personal Notes

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Table of Contents

Check if any large number is divisible by 17 or not

C++

Java

Python 3

C#

PHP

Javascript

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