# Zuckerman number

Consider the integer 384. Multiplying its digits,

$$3\times 8\times 4=96$$ |

and

$$\frac{384}{96}=91.$$ |

When an integer is divisible by the product of its digits, it’s called a Zuckerman number. That is, given $m$ is the number of digits of $n$ and ${d}_{x}$ (for $x\le k$) is an integer of $n$,

$$\prod _{i=1}^{m}{d}_{i}|n$$ |

All 1-digit numbers and the base number itself are Zuckerman numbers.

It is possible for an integer to be divisible by its multiplicative digital root and yet not be a Zuckerman number because it doesn’t divide its first digit product evenly (for example, 1728 in base 10 has multiplicative digital root 2 but is not divisible by $1\times 7\times 2\times 8=112$). The reverse is also possible (for example, 384 is divisible by 96, as shown above, but clearly not by its multiplicative digital root 0).

## References

- 1 J. J. Tattersall, Elementary number theory in nine chapters, p. 86. Cambridge: Cambridge University Press (2005)

Title | Zuckerman number |
---|---|

Canonical name | ZuckermanNumber |

Date of creation | 2013-03-22 16:04:36 |

Last modified on | 2013-03-22 16:04:36 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 4 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A63 |