2016 AMC 10B Problem 25

Let f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor), where \lfloor r \rfloor denotes the greatest integer less than or equal to r. How many distinct values does f(x) assume for x \ge 0?

\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}

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