### Instruction: How many zeroes does $10!$ end with, when $10!$ is written in base 9?### Response: To answer this question, I need to think about what it means to end with a zero in base 9.
In base 10, a zero at the end of a number means that it is divisible by 10, or by 2 and 5.
In base 9, a zero at the end of a number means that it is divisible by 9, or by 3 and 3.
So, I want to count how many factors of 3 there are in $10!$, since each pair of 3's will give me one zero in base 9.
To do this, I can use the formula for the highest power of a prime $p$ that divides $n!$, which is $\sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
In this case, $p = 3$ and $n = 10$, so I get $\sum_{k=1}^{\infty} \lfloor \frac{10}{3^k} \rfloor = \lfloor \frac{10}{3} \rfloor + \lfloor \frac{10}{9} \rfloor = 3 + 1 = 4$.
This means that there are four factors of 3 in $10!$, or two pairs of 3's.