0123456789 question

John proposes that the number $0123456789$ has the following rules:

1) If it is multiplied by a positive integer n, then the product uses the digits 0 to 9 only once.

2) If n is a multiple of 3 , then Rule 1 does not apply . Otherwise, Rule 1 applies.

3)The product has at least 10 digits. Any product that has less than 10 digits will have 0s added to the front until it reaches 10 digits.

Because he's a jerk, you want to prove him wrong. So, you search for a counterexample to his proposal. You are finding the smallest possible integer that disproves his proposal.

Does such a integer exist? If it does, what is it? If it doesn't, answer 0.