Show that there are infinitely many positive real numbers $a$ which are not integers such that $a(a-3\{a\})$ is an integer (here $\{a\}$ denotes the fractional part of $a$).
I tried putting $a = [a]+\{a\}$ which resulted in proving $([a]+\{a\})([a]-2\{a\})$ as an integer. Distributing, I got $[a]^2-[a]\{a\}-2\{a\}^2$ out of which $[a]^2$ is clearly an integer which implies $2\{a\}^2+[a]\{a\}$ is an integer. As $0<\{a\}<1$, we have $0<2\{a\}^2+[a]\{a\}<2+[a]$. I got stuck at this point and couldn't think of anything to do.
Any and all help is greatly appreciated!