Sweeping Floors

The solutions present in the original problem that inspired me used pattern searching. Here I present a more systematic approach.

We first apply the division algorithm on $x=107q+r$ where $r<107$ . Plugging this into the equation gives: $\left\lfloor \frac {107q+r}{100}\right\rfloor=q$ which by definition of floor functions is equivalent to $q\le \frac {107q+r}{100}<q+1$ Multiplying $100$ to both sides are collectively subtract $100q$ gives $0\le 7q+r<100$

Therefore it now suffices to find the number of non-negative integer pair $(q,r)$ that satisfy the inequality above as well as the presumed restrictions $r<107$ , which is implied by the inequality. This is now a routine counting problem: $q$ can take values $0,1,2,...,14$ , counting the number of corresponding $r$ gives $\displaystyle \sum_{q=0}^{14} 100-7q=15\cdot 100-7\frac {14\cdot 15}{2}=\boxed {765}$

Just list all the $x$ s.I did it!:D