Question 380080
Note that there are 9 one-digit numbers, 90 two-digit numbers, 900 three-digit numbers, etc. in base 10.

The first 9 digits are used to number 1-9. Then, 180 digits are used to number pages 10-99. This is 189 digits total, leaving 1,643 digits left. 1,643/3 = 547 R2, so 547 three-digit numbers will be completed, then two more before Dr. Math is interrupted.

The 547th three-digit number is 646 (note that the set is inclusive), so she will write 646, 64_ so the next digit is 7.