
6. Find all real numbers $a$ such that $3<a<4$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$. For example $\{1.5\}=0.5 ;\{-3.4\}=0.6$.)


Solution: Let $a=3+f$, where $00$, we get

$$
f=\frac{-3+\sqrt{17}}{4}, \quad \frac{1}{2}, \quad \frac{-3+\sqrt{33}}{4}, \quad \frac{-3+\sqrt{41}}{4}
$$

Therefore $a$ takes the values:

$$
\begin{gathered}
a=3+\frac{-3+\sqrt{17}}{4}, 3 \frac{1}{2}, 3+\frac{-3+\sqrt{33}}{4}, 3+\frac{-3+\sqrt{41}}{4} . \\
-0
\end{gathered}
$$

