The number has nine (not necessarily distinct) decimal digits. The number is such that each of the nine -digit numbers formed by replacing just one of the digits is by the corresponding digit is divisible by . The number is related to in the same way: that is each of the nine numbers formed by replacing one of the by the corresponding is divisible by . Show that for each , is divisible by . (For example, if then may be or , since and are multiples of ).

I've attempt some experimentation but the numbers are too big to play around with...

how to do it?