### Instruction: Let $N$ be the largest integer whose square has exactly $3$ digits when written in base 9.

What is $N$, expressed in base 9?### Response: I want to find the largest three-digit number in base 9 and then take its square root, rounding down to the nearest integer.
The largest digit in base 9 is 8, so the largest three-digit number in base 9 is 888.
I can rewrite 888 in base 9 as $8 \times 9^2 + 8 \times 9 + 8$.
Expanding and simplifying, I get $648 + 72 + 8 = 728$ in base 10.
Taking the square root of 728, I get about 26.981 in base 10.
Rounding down to the nearest integer, I get 26 in base 10.
To convert 26 to base 9, I divide by 9 and write the remainder as the last digit, then repeat with the quotient until I get 0.
26 divided by 9 is 2 with remainder 8, so the last digit is 8.
2 divided by 9 is 0 with remainder 2, so the next digit is 2.
There is no more quotient, so I stop and write the digits from right to left.
Therefore, 26 in base 10 is 28 in base 9.