A group of $10$ explorers agree to divide a treasure chest of silver coins among themselves as follows. The $k^{\text{th}}$ explorer to take a share takes $\frac{k}{10}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each explorer to receive a positive whole number of coins. How many coins does the $10^{\text{th}}$ explorer receive?