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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.

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