I'm thinking of a ten-digit integer whose digits are all distinct. it happens that the number formed by the first n of them is divisible by n for each n

The problem here takes a brilliant mind to answer this. This problem can easily be answered using programming because we can not then and there push all the possibilities using paper and pen.

The answer is 3816547290. 

Trying all the possibilities starting form 1000000080 (we are sure that the last number should be 0). Then traversing that number until 9999999990. Each traverse, check the number if its divisible to n, so on and so forth.