AMC 12 Mock Exam 2 Problem 5

There are $\lfloor200/6\rfloor=33$ multiples of $6$, $\lfloor200/15\rfloor=13$ multiples of $15$, and $\lfloor200/30\rfloor=6$ multiples of both (that is, of $30$), so $33+13-6=40$ numbers are divisible by $6$ or $15$.

Such a number that is also divisible by $10$ is divisible by $\operatorname{lcm}(6,10)=30$ or by $\operatorname{lcm}(15,10)=30$; that is, it is one of the $6$ multiples of $30$. Hence the answer is $40-6=\boxed{34}$.