A group of $8$ prospectors agree to divide a chest of gold nuggets among themselves as follows. The $k^{\text{th}}$ prospector to take a share takes $\frac{k}{8}$ of the nuggets that remain in the chest. The number of nuggets initially in the chest is the smallest number for which this arrangement will allow each prospector to receive a positive whole number of nuggets. How many nuggets does the $8^{\text{th}}$ prospector receive?