Let the emf of each identical cell be $E$ and internal resistance of each cell be $r$.

When the two cells are connected in series:

• Equivalent emf $= 2E$
• Equivalent internal resistance $= 2r$
• External resistance $R = 5\,\Omega$

So current in the external resistance is $I_s = \frac{2E}{R+2r} = \frac{2E}{5+2r}$

When the two cells are connected in parallel:

• Equivalent emf $= E$
• Equivalent internal resistance $= \frac{r}{2}$

So current in the external resistance is $I_p = \frac{E}{R+\frac{r}{2}} = \frac{E}{5+\frac{r}{2}}$

Given that both currents are the same: $I_s = I_p$ $\frac{2E}{5+2r} = \frac{E}{5+\frac{r}{2}}$

Cancel $E$ from both sides: $\frac{2}{5+2r} = \frac{1}{5+\frac{r}{2}}$

Cross-multiply: $2\left(5+\frac{r}{2}\right) = 5+2r$

Simplify: $10 + r = 5 + 2r$ $r = 5$

Therefore, the internal resistance of each cell is $\boxed{5\,\Omega}$