Begin with the 200 digit number 98765432198765 ... 543210, where the digits 0-9 are repeated in reverse order. From the left, choose every third digit to form a new number. Repeat the process with the new number. Continue repeatedly until the result is a two-digit number. What is the remaining two-digit number? (From Art of Problem Solving Introduction to Number Theory textbook, tried to solve but solution was not clear)

Answer:98

Step-by-step explanation:

We can begin by listing the numbers. We get, 74185 ... Then, we see that if the original list has 200 numbers, then the next list would have 200/3 numbers, then the next would have 200/9, and so on and so forth. Then, our 2nd list would be like this:3rd number, 6th number, 9th number, etc. Then our 3rd list would be : 9th number, 18th number, 27th number ... Then, our fourth number would be 27th number, 54th number, 81th number ... Thus, our last list would be 81th number, 162th number. So now, our problem is simply to find the 81th number and the 162th number, which is quite simple since it is just mod 10, cause it repeats every 10. So, the 81st number is 9 and the 162th number is 8. Thus, our answer is 98!