Zuckerman number

Consider the integer 384. Multiplying its digits,

3\times 8\times 4=96

and

{{384}\over{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 $x\leq k$ ) 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