Floor Functions

Let integral part of $x$ be $I$ and fractional part be $f$ .

$\lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 4x\rfloor+\lfloor 8x\rfloor+\lfloor 16x\rfloor+\lfloor 32x\rfloor=12345$ $\Rightarrow 63I+\lfloor 2f\rfloor+\lfloor 4f\rfloor+\lfloor 8f\rfloor+\lfloor 16f\rfloor+\lfloor 32f\rfloor=12345$

$12345\equiv 60\mod 63$

$\lfloor 2f\rfloor+\lfloor 4f\rfloor+\lfloor 8f\rfloor+\lfloor 16f\rfloor+\lfloor 32f\rfloor=60$

Note that maximum value of $\lfloor 2f\rfloor+\lfloor 4f\rfloor+\lfloor 8f\rfloor+\lfloor 16f\rfloor+\lfloor 32f\rfloor$ is $57$ when $f\rightarrow 1^{-}$ .

So no solution!

Properties used

• $\lfloor x+n\rfloor=n+\lfloor x\rfloor$ if n is integer.

A another way to guess : say it satisfies for one n then it would satisfy for another 54 values ( till floor(32x) increases by one) also ( making a total of 55 values) , so the answer could be either 55 if such value exist or zero