Editing Harshad number (section)

== Definition ==

Stated mathematically, let {{mvar|X}} be a positive integer with {{mvar|m}} digits when written in base {{mvar|n}}, and let the digits be <math>a_i</math> (<math>i = 0, 1, \ldots, m-1</math>). (It follows that <math>a_i</math> must be either zero or a positive integer up to {{tmath|n-1}}.) {{mvar|X}} can be expressed as

:<math>X=\sum_{i=0}^{m-1} a_i n^i.</math>

{{mvar|X}} is a harshad number in base {{mvar|n}} if:

:<math>X \equiv 0 \bmod {\sum_{i=0}^{m-1} a_i}.</math>

{{anchor|all-harshad number}}
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 (number)|1]], [[2 (number)|2]], [[4 (number)|4]], and [[6 (number)|6]]. The number [[12 (number)|12]] is a harshad number in all bases except [[octal]].