How many positive three-digit integers with a $5$ in the units place are divisible by $15$?
Right, a number is divisible by $5$ iff its units digit is $0$ or $5$. That's correct. Now we need to find out how many of those numbers are divisible by $15$. A positive integer is divisible by $15$ iff it is divisible by both $5$ and $3$. Right. Since we already know it is divisible by $5$, we just need to check for divisibility by $3$. A number is divisible by $3$ iff the sum of its digits is divisible by $3$. Exactly. Now we can solve the problem. The smallest positive three-digit integer with a $5$ in the units place is $105$. The smallest such integer is $105$. So we just need to find how many multiples of $3$ are between $0$ and $89$ inclusive. We can do that by dividing $89$ by $3$. We get $89/3=29$ with a remainder of $2$. Therefore, there are $30$ positive three-digit integers with a $5$ in the units place that are divisible by $15$. Right. So the answer is $30$.

