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Another way of writing $\lfloor x\rfloor = 4$ defines x values in the interval of $4 \leq x < 5$

If we multiply this interval by 4, we have that $16\leq 4x < 20$.

In order to get the sum, we have to acknowledge that our 4x is still being floored, given $\lfloor 4x \rfloor$. Therefore, only integer values within the interval can be a result of this. For example, $x = 4.625$ satisfies that $\lfloor x \rfloor = 4$. Then, $\lfloor 4x \rfloor = \lfloor 4*4.625\rfloor = \lfloor 18.5 \rfloor = 18$

The integers within $[16, 20)$ are 16, 17, 18, and 19. $16+17+18+19= 70$.

If $\lfloor x \rfloor = 4,$ the sum of all possible values of $\lfloor 4x \rfloor$ is $70$.