Let the initial number of atoms at time $t=0 \mathrm{b}\mathrm{e}\mathrm{ }{N}_0.$

Let $N$ be the number of atoms at any instant t.

Mean life $\tau =\frac{1}{\lambda }$ , where $\lambda$ is disintegration constant.

Given , $t= \tau$

According to radioactive disintegration law,

$N=N_0{e}^{-\lambda t}$

$\mathrm{o}\mathrm{r}\mathrm{ } N=N_0{e}^{-\lambda \times \frac{1}{\lambda }} = \frac{N_0}{e}$

$\mathrm{o}\mathrm{r} \frac{N_0}{N} = e$