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).