Can you solve the puzzle with explanation?

Question

There is a nine-digit number with each of the digits from 1 to 9 appearing once. The whole number is divisible by 9. If you remove the rightmost number, the remaining 8-digit number is divisible by 8. Remove the rightmost number again and the remaining 7-digit number is divisible by 7. The pattern continues right down to a single digit. What is the number?

Answer 1

First, any 9 digit number having all 9 digits is divisible by 9...

However,
381654729 is divisible by 9 (= 42406081)
38165472 is divisible by 8 (= 4770684)
3816547 is divisible by 7 (= 545221)
381654 is divisible by 6 (= 63609)
38165 is divisible by 5 (= 7633)
3816 is divisible by 4 (= 954)
381 is divisible by 3 (= 127)
38 is divisible by 2 (= 19) and...
3 of course is divisible by 1 :-P

Enjoy the extra credit :-O

Answer 2

The test for 1 is passed "for free".

The test for 9 is also passed "for free", since the sum of the digits is guaranteed to be divisible by 9.

The test for 5 tells us the 5th digit from the left has to be 5.

The tests for 2, 4, 6, and 8 tell us, among other things, that the 2nd, 4th, 6th, and 8th digits from the left are even. Those are all the even digits available, so the other five digits are odd.

The test for 3 tells us the three leftmost digits, added together, are divisible by 3. The same is true of the six leftmost digits, and hence of the middle three digits and the rightmost three digits.

The above tells us that the 4th and 6th digits are either 4,6 in some order or 2,8 in some order. The other pair comprises the 2nd and 8th digits.

Actually, the 4th digit from the left has to be 2 or 6, based on the test for 4 and the fact that the 3rd digit from the left is odd.

Not fully used to this point are the tests for 7 and 8.

The 2nd digit from the right can't be 8, because if it were, the 2-digit number formed immediately to its left would have to be divisible by 8, and in fact it's not even -- er, it's odd.

The rest of the solution looks a bit tedious. There are three remaining possibilities for the arrangement of the even digits, each of this will mandate 1 or at most two possibilities for the 3rd digit from the right (because of the test for 8). From there, using the fact about the first three digits and the last three digits each having their sums divisible by three will sharply limit the possibilities for the arrangement of the odd digits.

And to the limited number of possibilities thus left, you can apply the test for 7 to see what choices, if any, there are for the final answer.

Answer 3

First, any 9 digit number having all 9 digits is divisible by 9...

However,
381654729 is divisible by 9 (= 42406081)
38165472 is divisible by 8 (= 4770684)
3816547 is divisible by 7 (= 545221)
381654 is divisible by 6 (= 63609)
38165 is divisible by 5 (= 7633)
3816 is divisible by 4 (= 954)
381 is divisible by 3 (= 127)
38 is divisible by 2 (= 19)
3 is divisible by 1

Answer 4

381654729

Answer 5

381654729

Answer 6

This sounds like homework/bonus/extra credit.
Do it yourself.

Answer 7

I think the answer is 123456789