How many distinct values does $f(x)$ assume for $x \ge 0$

Question

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 $x \ge 0$?

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I first started out this problem with the memory that once a math teacher told me that $x = \lfloor x \rfloor + \{ x \}$ where $\{ x \}$ is the fractional part of $x$. I plug this in and get we have

$$f(x) = \sum_{k=2}^{10} (\lfloor k \lfloor x \rfloor +k \{ x \} \rfloor – k \lfloor x \rfloor)$$

The function can then be simplified into

$$f(x) = \sum_{k=2}^{10} ( k \lfloor x \rfloor + \lfloor k \{ x \} \rfloor – k \lfloor x \rfloor)$$

which becomes

$$f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor$$
I don't know how to continue.

Am I on the right track? If so, how should I continue? If not, what path should I take?

Also, if you are nice, could you please also help me on this problem($N$'s base-5 and base-6 representations, treated as base-10, yield sum $S$. For which $N$ are $S$'s rightmost two digits the same as $2N$'s?)?

Thanks!

Max0815

Answer 1

I solved it. Yay thanks Jens S. for helping me!