There are $\left\lfloor{\frac{2020}{3}}\right\rfloor =673$ numbers which are multiples of $3$, $\left\lfloor{\frac{2020}{4}}\right\rfloor =505$ numbers which are multiples of $4$, and $\left\lfloor{\frac{2020}{3\times 4}}\right\rfloor =168$ numbers which are multiples of $12$ (least common multiple of $3$ and $4$). Therefore there are $$673+505-168 =1 1010$$

numbers which are multiple of $3$ or $5$.

Meanwhile, therefore $\left\lfloor{\frac{2020}{3\times 4\times 5}}\right\rfloor = 33$ numbers which are multiple of $60$ (least common multiple of $3$, $4$, and $5$), therefore the final answer is $$1010-60=\boxed{950}$$