Find value of a in following number theory question How do I find all real numbers a such that 4 < a < 5 and  a(a-3{a}) is an integer. here {a} is an fractional part of a.

Letting $a=4+x$, where $x \in (0,1)$, we need $f(x) = (4+x)(4+x-3x) = (4+x)(4-2x)$ to be an integer. Rewriting we obtain
\begin{align}
f(x) & = 2(4+x)(2-x) \in \mathbb{Z}
\end{align}
Further on the interval $(0,1)$, we have $f(x)$ to be decreasing from $16$ to $10$. Since $f(x)$ is continuous, the integer values that $f(x)$ takes are $11,12,13,14$ and $15$. Each of this gives a value of $x$ from which you can find the value of $a$.
