Find the positive solutions for $x + 2 \{ x \} = 3 \lfloor x \rfloor$
attempt:
Notice that the equation can be rewritten as
$$ x + 2 \{ x\} = 2 \lfloor x \rfloor + x – \{x\}$$
$$ 3 \{x\} = 2 \lfloor x \rfloor $$
From this we know that $3 \{x\}$ must be even positive integer. But the only $\{x\}$ that makes it integer is $\{ x\} = 1/3$ (but $3\{x\}$ is odd) and $\{x\} = 0$. So there is no positive integer solution?
Best Answer
Hint:
You forgot about $\{x\}=2/3$.