Solution

In order that the pressure of gas in a container with constant volume $V_0$ and temperature $T_0$ increases from atmospheric pressure $P_0$ to $P$, the number of moles of gas molecules must increase by $(P-P_0)V_0/(RT_0)$. Each stroke of the pump injects gas of volume $V$ at atmospheric pressure $P_0$ and temperature $T_0$, which has $P_0V/(RT_0)$ moles of gas molecules, into the container. Thus, the number of strokes required is $(P-P_0)V_0/(P_0V)$.