Editing Harshad number (section)

== Properties ==

Given the [[divisibility test]] for 9, one might be tempted to generalize that all numbers divisible by 9 are also harshad numbers. But for the purpose of determining the harshadness of {{mvar|n}}, the digits of {{mvar|n}} can only be added up once and {{mvar|n}} must be divisible by that sum; otherwise, it is not a harshad number. For example, [[99 (number)|99]] is not a harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.

The base number (and furthermore, its powers) will always be a harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1.

All numbers whose base ''b'' digit sum divides ''b''−1 are harshad numbers in base ''b''.

For a [[prime number]] to also be a harshad number it must be less than or equal to the base number, otherwise the digits of the prime will add up to a number that is more than 1, but less than the prime, and will not be divisible. For example: 11 is not harshad in base 10 because the sum of its digits “11” is 1 + 1 = 2, and 11 is not divisible by 2; while in [[base 12]] the number 11 may be represented as “{{d3}}”, the sum of whose digits is also {{d3}}. Since {{d3}} is divisible by itself, it is harshad in base 12.

Although the sequence of [[factorial]]s starts with harshad numbers in base 10, not all factorials are harshad numbers. 432! is the first that is not. (432! has digit sum 3897 = 3<sup>2</sup> × 433 in base 10, thus not dividing 432!)

The smallest {{mvar|k}} such that <math>k \cdot n</math> is a harshad number are
:1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 9, 3, 2, 3, 6, 1, 6, 1, 1, 5, 9, 1, 2, 6, 1, 3, 9, 1, 12, 6, 4, 3, 2, 1, 3, 3, 3, 1, 10, 1, 12, 3, 1, 5, 9, 1, 8, 1, 2, 3, 18, 1, 2, 2, 2, 9, 9, 1, 12, 6, 1, 3, 3, 2, 3, 3, 3, 1, 18, 1, 7, 3, 2, 2, 4, 2, 9, 1, ... {{OEIS|id=A144261}}.

The smallest {{mvar|k}} such that <math>k \cdot n</math> is not a harshad number are
:11, 7, 5, 4, 3, 11, 2, 2, 11, 13, 1, 8, 1, 1, 1, 1, 1, 161, 1, 8, 5, 1, 1, 4, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 83, 1, 1, 1, 4, 1, 4, 1, 1, 11, 1, 1, 2, 1, 5, 1, 1, 1, 537, 1, 1, 1, 1, 1, 83, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 68, 1, 1, 1, 1, 1, 1, 1, 2, ... {{OEIS|id=A144262}}.