N is a positive base ten integer having at least 2-digits but at most 4-digits, which is obtained by multiplying the sum of its digits with the product of its digits. It is known that N cannot contain any leading zero.

Determine all possible value(s) of N.

The puzzle states that N cannot contain any LEADING zero. Can it have zero in any other position (if so, the product of those digits will be zero, and the overall multiplication will also be zero)? Perhaps we should just count a single value of zero once, and then inspect those where no digits are zero.