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# Harshad number

When an integer is divisible by the sum of its digits, it’s called a Harshad number or Niven number. That is, given m is the number of digits of n and d is an integer of n,

${\sum_{{i=1}}^{m}d_{i}}|n$ |

All 1-digit numbers and the base number itself are Harshad numbers. 1, 2, 4 and 6 are always Harshad numbers regardless of the base.

It is possible for an integer to be divisible by its digital root and yet not be a Harshad number because it doesn’t divide its first digit sum evenly (for example, 38 in base 10 has digital root 2 but is not divisible by 3 + 8 = 11). The reverse is also possible (for example, 195 is divisible by 1 + 9 + 5 = 15, but not by its digital root 4).

Defines:

Harshad number

Synonym:

Niven number

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11A63*no label found*

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