Question 19

We call an integer special if it is positive and we do not need to use the digit 0 to write it down in base 10. For example, 2126 is special whereas 2025 is not. The first 10 special numbers are $$1, 2, 3, 4, 5, 6, 7, 8, 9, 11$$. The 2025th special number is


Correct Answer: 2689

There are $$9 + 9^2 + 9^3 = 819$$ special numbers with at most three digits, so the required number is the $$2025 - 819 = 1206$$th four digit special number. Writing $$1206 - 1 = 1205$$ in base 9 gives the digits $$1, 5, 7, 8$$, and adding 1 to each digit converts it to the special number. This gives 2689.

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