# Definition:Niven Number

## Definition

A Niven number (in a given number base $b$) is a positive integer which is divisible by the sum of its digits in that given base $b$.

That is, $N$ is a Niven number base $b$ if and only if:

$\ds \exists A \in \Z: N = \sum_{k \mathop = 0}^m r_k b^k = A \sum_{k \mathop = 0}^m r_k$

where $\ds \sum_{k \mathop = 0}^m r_k b^k$ is the representation of $N$ in base $b$ as defined according to the Basis Representation Theorem.

## Source of Name

This entry was named for Ivan Morton Niven.