Ceilings? Floors?

Number Theory Level 3

$\left \lceil x \right \rceil + \left \lfloor x \right \rfloor + \left \{ x \right \} = 6$

How many different values for $x$ are there?

Clarification:

1 0 An infinite number 3 2 4 None of these answers

1 solution

Geoff Pilling
Mar 16, 2017

Since $\left \lceil x \right \rceil$ and $\left \lfloor x \right \rfloor$ are integers,

$\left \{ x \right \} = 0 \implies x$ is an integer.

Therefore,

$\left \lceil x \right \rceil + \left \lfloor x \right \rfloor = 6$

In general, for integer $x$ , $\left \lceil x \right \rceil = \left \lfloor x \right \rfloor$ , so:

$\left \lceil x \right \rceil + \left \lfloor x \right \rfloor + \left \{ x \right \} = 2x$

$\implies 2x = 6$

So, only $\boxed1$ possible answer exists, and that is $x=3$ .