Harshadnumber

A Harshad number, also known as a Niven number, is an integer that is divisible by the sum of its digits. This property is derived from Sanskrit, where "harshad" means "great joy". The concept was introduced by the Indian mathematician D. R. Kaprekar in 1956.

For example, the number 18 is a Harshad number because the sum of its digits is 1 + 8 = 9, and 18 is divisible by 9 (18 / 9 = 2). Similarly, 1729 is a Harshad number as its digits sum to 1 + 7 + 2 + 9 = 19, and 1729 is divisible by 19 (1729 / 19 = 91).

All single-digit numbers (1 through 9) are Harshad numbers since any single digit is divisible by itself. However, as numbers grow larger, the occurrences of Harshad numbers become less frequent. The property is base-dependent, meaning a number that is a Harshad number in one base might not be in another. For instance, 21 is a Harshad number in base 10 (2+1=3, 21/3=7), but in base 2, 21 is represented as 10101, the sum of its digits is 3, and 10101 (21) is not divisible by 3.

The study of Harshad numbers is a part of recreational mathematics and number theory. There are no known significant mathematical applications for Harshad numbers outside of this domain.