Harshad number

In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-Harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. All integers between zero and n are n-Harshad numbers.

Definition

Stated mathematically, let X be a positive integer with m digits when written in base n, and let the digits be ai (i = 0, 1, ..., m  1). (It follows that ai must be either zero or a positive integer up to n  1.) X can be expressed as

If there exists an integer A such that the following holds, then X is a Harshad number in base n:

A number which is a Harshad number in every number base is called an all-Harshad number, or an all-Niven number. There are only four all-Harshad numbers: 1, 2, 4, and 6 (The number 12 is a Harshad number in all bases except octal).

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Kumkum Bhagya: Pranali Rathod on challenges of portraying Prarthana, shares how she is preparing for the role

Bollywood Life 11 Mar 2025
The TRP numbers are looking great too ... Rajan Shahi reveals Harshad Chopda wanted to be a part of 'number one' show, 'Sad part was...' ... Also Read - Yeh Rishta Kya Kehlata Hai's Pranali Rathod fought with Harshad Chopda and deleted their photos?.
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Yeh Rishta Kya Kehlata Hai: Samridhii Shukla, Rohit Purohit perform on Ranveer Singh, Deepika Padukone’s song; fans say ‘Too good to…’

Bollywood Life 11 Mar 2025
By Sanskruti Nemane Published. March 11, 2025 10.32 AM IST.  Follow Us ... The story of Abhira and Armaan has grabbed attention ... Rajan Shahi reveals Harshad Chopda wanted to be a part of 'number one' show, 'Sad part was...' Abhira, Armaan's dance ... .
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Yeh Rishta Kya Kehlata Hai: Rohit Purohit reacts to the show being copied from Fahad Mustafa, Hania Aamir's Kabhi Main Kabhi Tum

Bollywood Life 11 Mar 2025
By Sanskruti Nemane Published. March 11, 2025 6.00 AM IST.  Follow Us ... They play the role of Abhira and Armaan in the serial ... Rajan Shahi reveals Harshad Chopda wanted to be a part of 'number one' show, 'Sad part was...' ... .
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Yeh Rishta Kya Kehlata Hai: Rajan Shahi reveals Harshad Chopda wanted to be a part of 'number one' show, 'Sad part was...'

Bollywood Life 10 Mar 2025
Recently, Rajan Shahi spoke about Harshad Chopda and mentioned that the actor desired to be part of a number show ... He recalled how Harshad called him once and said that he wants to work in a number one show.
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Yeh Rishta Kya Kehlata Hai star Garvita Sadhwani finds her character Ruhi 'powerful'; says 'She runs high on...'

Bollywood Life 10 Mar 2025
Garvita Sadhwani came onboard Yeh Rishta Kya Kehlata Hai after Pratiksha Honmukhe's ouster. She plays the role of Ruhi in the show ...  Follow Us ... Rajan Shahi reveals Harshad Chopda wanted to be a part of 'number one' show, 'Sad part was...' ... .
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