May you log long

Let $x = \log_{10}15$ and $y = \log_{10}16$ and set $A= \{ \log_{10}1 , \log_{10}2 , \ldots , \log_{10}50 \}$ . Find the number of elements in set $A$ that can be written in the form of $ax+by+c$ , where $a$ , $b$ , and $c$ are rational numbers .