# Definition:Niven Number

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## 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:

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

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

## Also see

## Source of Name

This entry was named for Ivan Morton Niven.