The initial and final amount of the radioactive substance are related by the formula

$\frac{N}{N_0}={\left(\frac{1}{2}\right)}^n ...\left(1\right)$

where, $n$ is the number of half lives.

Substitute $\frac{1}{8}$ for $\frac{N}{N_0}$ into equation (1) and solve to calculate the value of $n$.

$\begin{array}{rcl}\frac{1}{8}& =& {\left(\frac{1}{2}\right)}^n\\ & \Rightarrow & {\left(\frac{1}{2}\right)}^3={\left(\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)}^n\\ & \Rightarrow & n=3\end{array}$

The half-life of the radioactive element can be calculated as follows:

$\frac{t}{t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}=n ...\left(2\right)$

Substitute the values of the known parameters into equation (2) and solve to calculate the half-life.

$\begin{array}{rcl}\frac{3 \mathrm{days}}{t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}& =& 3\\ & \Rightarrow & t_{\raisebox{1ex}{${\displaystyle 1}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle 2}$}\right.}= 1 \mathrm{day}\end{array}$

The initial and the final mass of the radioactive element are related by

$\frac{m}{m_0}={\left(\frac{1}{2}\right)}^{\frac{t\text{'}}{t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}} ...\left(3\right)$

Substitute the values of the known parameters into equation (3) and solve to calculate the initial mass of the substance.

$\begin{array}{rcl}\frac{8\times {10}^{-3} \mathrm{kg}}{m_0}& =& {\left(\frac{1}{2}\right)}^{\frac{5}{1}}\\ & \Rightarrow & m_0= 2^5\times 8\times {10}^{-3} \mathrm{kg}\\ & =& 256 \mathrm{g}\end{array}$