Prove $\alpha f + \beta g$ is periodic function with period LCM($\tau_1,\tau_2$).

Question

Let $f$ be periodic with period $\tau_1$ and $g$ periodic with period $\tau_2$, such that $\frac{\tau_1}{\tau_2} \in \mathbb{Q}$. Then prove $\alpha f + \beta g$ is periodic function with period LCM($\tau_1,\tau_2$).

I know what is intuitive idea why this works,but I have problem with trying to prove it. The biggest problem is question where to use $\frac{\tau_1}{\tau_2} \in \mathbb{Q}$?