solution: We could do this by the divisibility rules, but that would be quite tedious. It's easier to note that a number divisible by $3, 4,$ and $5$ must be divisible by their product, $3 \times 4 \times 5 = 60$. This is because a number which is divisible by several integers must be divisible by their least common multiple -- however, since $3, 4,$ and $5$ are relatively prime, the least common multiple is just the product of all three. Clearly, there is only one number between $1$ and $100$ divisible by $60;$ that is, $60$ itself. Thus there is only $\boxed{1}$ such number.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  How many of the first one hundred positive integers are divisible by $3, 4,$ and $5?$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 1
extra_info.question: How many of the first one hundred positive integers are divisible by $3, 4,$ and $5?$
extra_info.split: test