Floor function of product

Question

We have real number $a$ that satisfies $floor(ax)=afloor(x)$ for every $x$ real number. We have to show that $a$ is an integer. So far I've been able to show that $a$ is a positive rational number.

Answer 1

Hint: If $a$ is rational I can say $a=\frac \ell b$ for $\ell,b$ are integers.

Try to prove (if you didn't already) that all integers have the property you said, so $floor(\ell x/b)=\ell floor(x/b)$ now try to prove that $b=1$

Answer 2

Hint:   for $\,x=1\,$ that reduces to $\,a = \lfloor a \rfloor\,$.