Harshad numbers - Ebhor.com

In this universe, we know that the number system is the base of every mathematical model. When we talk about programming, we do deal with numbers and a way to manipulate the data variables through in-built functions and modules.

Lots of concepts are there from which one is a Harshad Number.

Another name of Harshad numbers are Niven numbers.

Niven numbers in base n are also called n-Niven ( or n-Harshad) numbers.

The word “harshad” comes from the Sanskrit harṣa (joy) + da (give) means joy-giver.

D. R. Kaprekar, a mathematician from India defined Harshad numbers.

The term “Niven number” arose by Ivan M. Niven.

Definition:- Harshad number is the smallest integer number in a given number base which is divisible by the sum of its digits when written in that base is known as Harshad number.

Steps To Find A Harshad Number

Firstly, we need to accept the data value as input.
Then we need to segregate the original number to get hold of all the digits that make up the number individually.
Then, we do the sum of all the numbers that make up the required number.
The last and final step is to divide the original number by the summation score.
If we get any non-zero value as the output of the modulus operation, we can state that the original number is not a Harshad number.
If the result is 0, it implies that the number is completely divisible by the summation score. Thus, we can conclude that an original number is a Harshad number.

Now let us take an example to understand the concept in a better way :-

Input number: 126

Digits of the number: 1, 2 and 6

Summation of digits: 1 + 2 + 6 = 9

Modulus operation: 126% 12 = 0

As the result of the mod, the operation is zero, we state that 126 is a Harshad number.

Now take some more examples:-

If a number is 18, the sum of the digits is 1+ 8 = 9 and the number 18 is divisible by 9. So 18 is a Harshad number.
If a number is 29, the sum of the digits is 2+ 9 = 11 and the number 29 is not divisible by 11. So 29 is not a Harshad number.
Some of the Harshad numbers represented in base 10 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30,………
The Hardy – Ramanujan number i.e. 1729 is a Niven number in base 10 as it is divisible by ( 1 + 7 + 2 + 9) 19.

Note :-The most popular discovery remains the Hardy-Ramanujan number to this date. Hardy, the British mathematician said that he came in a taxi having the number ‘1729’to meet S Ramanujan in hospital which the British mathematician described “as rather a dull one”. Ramanujan replied to this saying, “No Hardy, it’s a very interesting number! It can be expressed as a sum of two positive integer cubes in n distinct ways i.e. 9³ + 10³ and 1³ + 12³. So, it is also known as the taxicab number.

Super Niven Number

If a number N not only divisible by the sum of its digits but also by the sum of any subset of its (nonzero) digits is known as Super Niven Number.

For Example:

68040

Divisible by 6, 8, 4, 6+8, 6+4, 4+8 and 6+4+8 so it is a Super Niven Number.

All-Harshad Number

A number that is a Harshad number is also known as an all- Harshad number in any number base.

Multiple Harshad Number

In 2005, Bloem said a Harshad number is known as a multiple Harshad number ( MHN) if a number divided by the sum of its digits produces another harshad number.

For Example:

6804

6804 ÷ 18 = 378

378 ÷ 18 = 21

21 ÷ 3 = 7

7 ÷ 7 = 1

Hence, 6804 is “MHN-4”

Note :- 6804 is not MHN-5 because1/1 = 1 (is not “another” harshad number)

Super-Harshad

If a number is divisible by its sum of digits s, and furthermore, the quotient is none other than the number s with its digits reversed.

For Example:- 1729

the sum of digits = 1 + 7 + 2 + 9 = 19 ,and indeed 1729=19×91 , with 91 being the number 19 with digits reversed.

Points To Remember

All 1-digit numbers are Harshad numbers.
1, 2, 4, and 6 are only four all-Harshad numbers.
The number 12 in all bases except octal is a Harshad number.

Properties Of A Harshad Number

The base number and furthermore, its powers will always be a harshad number in its own base, since it will be represented as “10” and 1 + 0 = 1.
All numbers whose base b digit sum divides b−1 are harshad numbers in base b.
Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also harshad numbers but the digits of n can only be added up once and n must be divisible by that sum for the purpose of determining the harshadness of n. So it is not a harshad number. For example, 99 is not a harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.
If a prime number is less than or equal to the base number then it is also a Harshad number Else the digits of the prime no will sum up to a no that is greater than one, but less than the prime no, and can not be divided. For example:Take a number 11,the sum of its digits = 1 + 1 = 2 and 11 is not divisible by 2. So, 11 is not harshad number in base 10.
All factorials are not harshad numbers although the sequence of factorials starts with harshad numbers in base 10.Example: 432! Is the first that is not. (432! Has digit sum = 3897 = 32×433 in base 10, thus not dividing 432!)

Algorithm to determine a Harshad number

We need to follow the below algorithm to check for a Harshad Number.

Accept the number as input.
By storing the value of num in variable n, make a copy of the num.
Calculate the remainder by dividing num with 10, if num is greater than 0.
Addition of rem to a variable sum.
Divide num by 10.
To calculate the sum of all digits present in a given number, repeat the steps from 3 to 5.
If n (copy of original number) is divisible by the sum, then given number is a Harshad number, otherwise given number is not a Harshad number.

Harshad Number Program In Java

import java.util.Scanner;
public class HarshadNumber{
     public static void main(String args[]){
        Scanner sc = new Scanner(System.in);
        System.out.print("Input a number : ");
        int num = sc.nextInt();
        int x = num, y, sum = 0;
      while(x&gt;0)
        {
            y = x%10;
            sum = sum + y;
            x = x/10;
        }
       if(num%sum == 0){
        System.out.println(num+ " is a Harshad Number.");
       }
        else{
            
             System.out.println(num+" is not a Harshad Number.");    
        }
    }
}

import java.util.Scanner;

public class HarshadNumber{

public static void main(String args[]){

Scanner sc = new Scanner(System.in);

System.out.print("Input a number : ");

int num = sc.nextInt();

int x = num, y, sum = 0;

while(x>0)

{

y = x%10;

sum = sum + y;

x = x/10;

}

if(num%sum == 0){

System.out.println(num+ " is a Harshad Number.");

}

else{

System.out.println(num+" is not a Harshad Number.");

}