More 10-Digit Numbers

• Find the smallest 10 digit number that uses all the digits 0-9, and the first N digits (taken as an N-digit number) are divisible by N.
• Now cycle the division requirement.  Find the smallest 10 digit number that uses all the digits 0-9, and the first N digits are divisible by (N+k mod 10 + 1) for k = 0 to 8.
(Sorry that expression looks worse than it is.)
• For example, for k=0, The first digit is divisible by 2, the first two digits are divisible by 3, ..., all 10 are divisible by 1.
• For k=7, The first digit is divisible by 9, the first two digits are divisible by 10, ..., all 10 are divisible by 8.

Source: Original, based on a previous puzzle from Martin Gardner.