Question 19
Let $A$ be the set of positive integers that have no prime factors other than $2,$ $3,$ or $5.$ The infinite sum $$
\begin{aligned} &\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\ &\quad {}+ \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} \\ &\quad {}+ \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} \\ &\quad {}+ \frac{1}{18} + \frac{1}{20} + \cdots \end{aligned}
$$ of the reciprocals of all the elements of $A$ can be expressed as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$
\begin{aligned} &\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\ &\quad {}+ \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} \\ &\quad {}+ \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} \\ &\quad {}+ \frac{1}{18} + \frac{1}{20} + \cdots \end{aligned}
$$ of the reciprocals of all the elements of $A$ can be expressed as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$