How many of the first 500 positive integers are divisible by 3, 4 and 5?
The least common multiple of 3, 4, and 5 is $3\cdot4\cdot5=60$, so we must find how many multiples of 60 are less than 500.  These multiples of 60 are $1\cdot 60, 2\cdot 60, \ldots 8\cdot 60$.  Thus, $\boxed{8}$ of the first 500 positive integers are divisible by 3, 4, and 5.