If a temperature of a reaction is raised from 300 to 320K, what factor will the reaction rate increase by
The amount of increase is dependent on the activation energy. For a very low activation energy, there will be almost no change (this is what you obtained by removing $E_a$ and $T$). For a high activation energy, the change will be quite large. For example, with an activation energy of $110\mathrm{\frac{kJ}{mol}}$, you the rate will increase by about $16$ times. $$\frac{\exp({\frac{-E_a}{R*320}})}{\exp({\frac{-E_a}{R*300}})}=\frac{\exp({\frac{-111000\text{J}}{8.314*320}})}{\exp({\frac{-111000\text{J}}{8.314*300}})}=\frac{7.6\times10^{-19}}{4.7\times10^{-20}}=16.14$$

So there really isn't an "in general" answer. However, many text books give a crude estimate that the rate will double when the temperature is increased by $10$. This isn't in general the case, as it depends on the temperature range (a change from $300$ to $320$ is more significant that a change from $600$ to $620$) and the the value of the activation energy.