### Instruction: How many zeroes does $10!$ end in, when written in base 11?### Response: I know that the number of zeroes at the end of a factorial in base 10 depends on how many factors of 10 there are in the factorial, which in turn depends on how many factors of 5 there are, since there are always more factors of 2 than 5.
For example, 10! in base 10 has two zeroes, because there are two factors of 5: 5 and 10.
So I want to find out how many factors of 11 there are in 10!, since 11 is the base I'm working with.
I know that 11 is a prime number, so the only way to get a factor of 11 is to have 11 itself as a factor.
But 11 is not a factor of 10!, since 10! is the product of all the positive integers from 1 to 10.