For how many integer values of $n$ between 1 and 474 inclusive does the decimal representation of $\frac{n}{475}$ terminate?
First, let's look at the prime factorization of $475$. It's $5^2 * 19$. So since $475$ has a prime factor of $19$, only $n$'s that are multiples of $19$ will terminate. The multiples of 19 between 1 and 474 inclusive are 19, 38, 57, ..., 454. We can use arithmetic sequences. The formula for the $n$th term is $19 + (n-1)*19$. And what is the largest $n$ such that $19n \leq 474$? That is 24. So there are 24 multiples of 19 between 1 and 474 inclusive.



