### Instruction: For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{1375}$ terminate?### Response: I know that a decimal representation of a fraction terminates if and only if the denominator has no prime factors other than 2 and 5.
So I want to factor 1375 and see what primes it has.
I notice that 1375 is divisible by 5, so I can write it as $5 \times 275$.
Then I see that 275 is also divisible by 5, so I can write it as $5 \times 5 \times 55$.
Finally, I realize that 55 is $5 \times 11$, so I have $1375 = 5^3 \times 11$.
This means that the only way for $\frac{n}{1375}$ to terminate is if $n$ is a multiple of 11.
There are 90 multiples of 11 between 1 and 1000, so that is the answer.