jmc

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.