A group of $10$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{th}$ pirate 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 pirate to receive a positive whole number of coins. How many coins does the $10^{th}$ pirate receive?