2018 AMC 12A Problem 19

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

\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 36

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.