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Product Divisibility Number Integer Base Divisibility In Rings Multiplicative Digital Root

1	Product
1	Integer
1	Number
1	Divisibility
1	Base
1	Divisibility In Rings
15	Multiplicative Digital Root

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Zuckerman Number

Consider the integer 384. Multiplying its digits,

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

and

$\displaystyle {{384} \over {96}} = 91.$

When an integer is divisible by the product of its digits, it's called a Zuckerman number. That is, given is the number of digits of and (for $x \le k$ ) is an integer of ,

$\displaystyle {\prod_{i = 1}^m d_i}\vert n$

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).

Bibliography

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