Check if any large number is divisible by 17 or not
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:
- 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 :
Letbe any number such that
=100a+10b+c .
Now assume thatis divisible by 17. Then
0 (mod 17)
100a+10b+c0 (mod 17)
10(10a+b)+c0 (mod 17)
10+c
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 of1.
In other words, we have to find an integer such that n such that 10n1 mod 17.
It can be observed that the smallest n which satisfies this property is -5 as -501 mod 17.
Now we can multiply the original equation 10+c
0 (mod 17)
by -5 and simplify it:
-50-5c
0 (mod 17)
-5c
0 (mod 17)
We have found out that if0 (mod 17) then,
-5c
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> |
Yes
Time Complexity: O(log10n), time required to check if number is divisible by 17
Auxiliary Space: O(1), as no extra space is required
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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