Write $x^2$ in two different ways and get $$\lfloor x^2 \rfloor + \{ x^2 \} = \lfloor x\rfloor^2 + 2\lfloor x \rfloor \{x\} + \{ x\}^2.$$ Since $\{ x^2 \} = \{x\}^2$, we get $$\lfloor x\rfloor^2 + 2\lfloor x \rfloor \{x\}.$$  Thus we know that $2\lfloor x \rfloor \{x\}$ must be an integer.  For $1\leq x < 7$, there are only a few possible values for $\lfloor x \rfloor$, and try each of them out.