Need help with American Billion problem

I need help finding a number, please post only clues and not an answer.

'Find a ten-digit number which uses each of the digits 0-9 once and has the following properties'

First digit is divisible by 1

Number formed by the first 2 digits from the left is divisible by 2

Number formed by the first 3 digits from the left is divisible by 3

(so on...up to 9)

Number itself is divisible by 10

I've worked out that there are 9 different possibilities for the first 3 numbers, I think 15 for the first 4 (please tell me if I'm wrong, but only give a clue not the answer) and 16 for the first 5.

I just have no idea how to incorporate the idea that although a number is a possibility for example statement 3 or 4, it may not be possible for the 7th statement (Number formed by the first 7 digits...). Nor how I can check these possibilities and see if all their numbers are different.

No calculator, (yes a spreadsheet would be great, but no :( ), and no A-level maths or higher.

I need to find a number but preferrably how many numbers that correspond to these rules exist.

Please help me out with clues.

Thanks.