Q:

What is this Number?

N is a number where each digit is divisible by its place, for example:

1527

1 has to be divisible by 1, 5 divisible by 2, 2 divisible by 3, and 7 divisible by 4.
Additionally, N must be divisible by all of its digits.
It has to be at least 5 digits long, and have at least 3 different digits.
Does N exist? Is there more than 1 number that fits N? If so is there a pattern? Give a few examples.

A:

 If such an N exists, it is at least 5 digits long, so it has a 2nd digit and a 5th digit.
 The second digit must be divisible by 2, and the 5th digit must be divisible by 5.
 N is divisible by both of these, so it must be divisible by 10.
 Then the last digit of N is 0.
 But no positive number can be divisible by 0, so N cannot exist.

