How many positive integers less than $250$ are multiples of $5$, but not multiples of $10$?
I need to find the numbers that are divisible by $5$, but not by $2$, because that would make them divisible by $10$ as well. One way to do this is to list all the multiples of $5$ less than $250$, and then eliminate the even ones. The multiples of $5$ less than $250$ are $5, 10, 15, 20, 25, ..., 240, 245$. There are $\frac{245 - 5}{5} + 1 = 49$ of them in total. The even ones are $10, 20, 30, ..., 240$, which are $5$ times $2, 4, 6, ..., 48$. There are $\frac{48 - 2}{2} + 1 = 24$ of them in total. So the number of odd multiples of $5$ less than $250$ is $49 - 24 = 25$. These are the numbers that are multiples of $5$, but not multiples of $10$. 