Let $x,N$ be positive integers.I'd like to store a counter which could reach value of $\left\lfloor \frac{N}{x}\right\rfloor$ (i.e. could take any value in $0,1,\ldots,\frac{N}{x}$) using  $$\lceil\log_2(N+1)\rceil - \lfloor\log_2 x\rfloor$$bits.Equivalently, does the following hold:$$\log\left\lfloor \frac{N}{x} + 1\right\rfloor \leq \lceil\log_2(N+1)\rceil - \lfloor\log_2 x\rfloor$$Obviously, this holds for $x=1$, what about general $x$?
Is it possble to store a counter that could reach $\lfloor \frac{N}{x}\rfloor$ using $\lceil\log_2(N+1)\rceil$ - $\lfloor\log_2 x\rfloor$ bits?
encoding scheme;space analysis