### Instruction: The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.### Response: I notice that $n$ must be a multiple of both $3$ and $5$, since $15 = 3 \times 5$.
To be a multiple of $3$, the sum of the digits of $n$ must be a multiple of $3$.
To be a multiple of $5$, the last digit of $n$ must be either $0$ or $5$.
Since every digit of $n$ is either $8$ or $0$, the last digit must be $0$, and the sum of the digits must be a multiple of $24$, since $8 + 8 + 8 = 24$.
The largest possible sum of the digits of $n$ is $240$, which occurs when $n$ has ten digits, all of them $8$ except the last one, which is $0$.
So $n = 8888888880$, and $\frac{n}{15} = \frac{8888888880}{15} = \frac{592592592}{1}$.