# 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).

Title | Harshad number |
---|---|

Canonical name | HarshadNumber |

Date of creation | 2013-03-22 15:47:04 |

Last modified on | 2013-03-22 15:47:04 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A63 |

Synonym | Niven number |

Defines | Harshad number |