1. How many positive integers less than 2019 are divisible by either 18 or 21 , but not both?
Solution: For any positive integer $n$, the number of multiples of $n$ less than or equal to 2019 is given by
$$
\left\lfloor\frac{2019}{n}\right\rfloor .
$$

So there are $\left\lfloor\frac{2019}{18}\right\rfloor=112$ multiples of 18 , and $\left\lfloor\frac{2019}{21}\right\rfloor=96$ multiples of 21 . Moreover, since $\operatorname{lcm}(18,21)=126$ there are $\left\lfloor\frac{2019}{126}\right\rfloor=16$ positive integers less than 2019 which are a multiple of both 18 and 21 . Therefore the final answer is
$$
\left\lfloor\frac{2019}{18}\right\rfloor+\left\lfloor\frac{2019}{21}\right\rfloor-2\left\lfloor\frac{2019}{126}\right\rfloor=112+96-2 \times 16=176 .
$$