### 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 need to find the largest integer $N$ such that $N^2$ has $3$ digits in base 9.
That means $N^2$ must be between $9^2$ and $9^3$, since $9^2$ has $2$ digits and $9^3$ has $4$ digits in base 9.
In decimal, that means $N^2$ must be between $81$ and $729$.
To find the largest possible $N$, I want to find the largest perfect square in that range.
I know that $25^2 = 625$, which is in the range, and $26^2 = 676$, which is also in the range, but $27^2 = 729$, which is not.
So the largest perfect square in the range is $26^2 = 676$.
Therefore, the largest possible $N$ is $26$.
To express $N$ in base 9, I need to write it as a sum of powers of 9.
I know that $26 = 2 \times 9 + 8$, so $N = 28$ in base 9.