Let's translate each of these sentences into equations: Letting and so on,
Summing the first equation over all i we get
Summing the first two equations gives
since 3 and 7 are relatively prime.
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?
Let's translate each of these sentences into equations: Letting and so on,
Summing the first equation over all i we get
Summing the first two equations gives
since 3 and 7 are relatively prime.
I finally get it ... I'm not sure how you got though (I'm assuming it's related to the mod) ... I would use instead ....
usagi, the first two congruences are basically the individual replacing of digits. You start with the number , then you replace one of it's digits by , such that it's divisible by 7. This is the same as adding the difference between and to the appropriate digit. Multiplying the difference by (instead of ) will take care of which digit the difference will be added to. Same thing for converting the number .
When you add together the first congruence for all (i.e. , you get the third congruence.
Can you figure it out from there?