# 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

Title Zuckerman number ZuckermanNumber 2013-03-22 16:04:36 2013-03-22 16:04:36 CompositeFan (12809) CompositeFan (12809) 4 CompositeFan (12809) Definition msc 11A63